Interactions between commuting individuals can lead to large-scale spreading of rumors, ideas, or disease, even though the commuters have no net displacement. The emergent dynamics depend crucially on the commuting distribution of a population, that is how the probability to travel to a destination decays with distance from home. Applying this idea to epidemics, we will demonstrate the qualitatively different infection dynamics emerging from populations with different commuting distributions. If the commuting distribution is exponentially localized, we recover a reaction-diffusion system and observe Fisher waves traveling at a speed proportional to the characteristic commuting distance. If the commuting distribution has a long tail, then no finite-velocity waves can form, but we show that, in some regimes, there is nontrivial spatial dependence that the well-mixed approximation neglects. We discuss how, in all cases, an initial dispersal-dominated regime can allow the disease to go undetected for a finite amount of time before exponential growth takes over. This "offset time" is a quantity of huge importance for epidemic surveillance and yet largely ignored in the literature.
In this paper we introduce a space-dependent multiscale model to describe the spatial spread of an infectious disease under uncertain data with particular interest in simulating the onset of the COVID-19 epidemic in Italy. While virus transmission is ruled by a SEIAR type compartmental model, within our approach the population is given by a sum of commuters moving on a extra-urban scale and non commuters interacting only on the smaller urban scale. A transport dynamic of the commuter population at large spatial scales, based on kinetic equations, is coupled with a diffusion model for non commu…
The importance of spatial networks in the spread of an epidemic is an essential aspect in modeling the dynamics of an infectious disease. Additionally, any realistic data-driven model must take into account the large uncertainty in the values reported by official sources, such as the amount of infectious individuals. In this paper we address the above aspects through a hyperbolic compartmental model on networks, in which nodes identify locations of interest, such as cities or regions, and arcs represent the ensemble of main mobility paths. The model describes the spatial movement and interacti…
In 2020 and 2021, the spread of COVID-19 was globally addressed by imposing restrictions on the distance of individual travel. Recent literature has uncovered a clear pattern in human mobility that underlies the complexity of urban mobility: $r \cdot f$, the product of distance traveled $r$ and frequency of return $f$ per user to a given location, is invariant across space. This paper asks whether the invariant $r\cdot f$ also serves as a driver for epidemic spread, so that the risk associated with human movement can be modeled by a unifying variable $r\cdot f$. We use two large-scale datasets…
Infectious diseases spread through human networks. Susceptible-Infected-Removed (SIR) model is one of the epidemic models to describe infection dynamics on a complex network connecting individuals. In the metapopulation SIR model, each node represents a population (group) which has many individuals. In this paper, we propose a modified metapopulation SIR model in which a latent period is taken into account. We call it SIIR model. We divide the infection period into two stages: an infected stage, which is the same as the previous model, and a seriously ill stage, in which individuals are infect…
Using a probability of novel encounter derived from a physical model, we augment the SIR compartmental model for disease spread. Scenarios with the same initial trajectories and identical $R_0$ values can diverge greatly depending on the speed at which our circles of acquaintances grow stale - leading to order of magnitude differences in final case counts. A momentum effect arises from variation in the mean time since infection, and this feeds back into new infection rate and faster decline in the late stages of an outbreak. Rapid extinction of an outbreak can occur in the early stages, but on…
In numerous contexts, individuals may decide whether they take actions to mitigate the spread of disease, or not. Mitigating the spread of disease requires an individual to change their routine behaviours to benefit others, resulting in a 'disease dilemma' similar to the seminal prisoner's dilemma. In the classical prisoner's dilemma, evolutionary game dynamics predict that all individuals evolve to 'defect.' We have discovered that when the rate of cooperation within a population is directly linked to the rate of spread of the disease, cooperation evolves under certain conditions. For disease…
The spread of certain diseases can be promoted, in some cases substantially, by prior infection with another disease. One example is that of HIV, whose immunosuppressant effects significantly increase the chances of infection with other pathogens. Such coinfection processes, when combined with nontrivial structure in the contact networks over which diseases spread, can lead to complex patterns of epidemiological behavior. Here we consider a mathematical model of two diseases spreading through a single population, where infection with one disease is dependent on prior infection with the other. …
This article is aimed at proposing a generalization of the Ross-Macdonald model for the transmission of Vector-borne diseases in which human-to-human contagions are also considered. We first present this generalized model by formulating a mean field theory, checking its validity by comparing to numerical simulations. To make the premises of our model more realistic, we adapt the mean field equations to the case in which human contacts are described by a complex network. In this case we are also able to derive an analytical expression for the epidemic threshold. In both the mean-field and netwo…
Modeling plant disease spread via high-resolution human mobility networks
Center for Complex Networks and Systems Research, Luddy School of Informatics, Computing, and Engineering, Indiana University（印第安纳大学卢迪信息、计算与工程学院复杂网络与系统研究中心）；Indiana University（印第安纳大学）；School of Public Health, Indiana University（印第安纳大学公共卫生学院）；Onside
Infectious diseases are a significant threat to human society which was over sighted before the incidence of COVID-19, although according to the report of the World Health Organisation (WHO) about 4.2 million people die annually due to infectious disease. Due to recent COVID-19 pandemic, more than 2 million people died during 2020 and 96.2 million people got affected by this devastating disease. Recent research shows that applying individual interactions and movements data could help managing the pandemic though modelling the spread of infectious diseases on social contact networks. Infectious…
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 trave…
In this paper, we develop a probabilistic mathematical model for the spread of coronavirus disease (COVID-19). It takes into account the known special characteristics of this disease such as the existence of infectious undetected cases and the different social and infectiousness conditions of infected people. In particular, it considers the social structure and governmental measures in a country, the fraction of detected cases over the real total infected cases, and the influx of undetected infected people from outside the borders. Although the model is simple and allows a reasonable identific…
Diseases and other contagion phenomena in nature and society can interact asymmetrically, such that one can benefit from the other, which in turn impairs the first, in analogy with predator-prey systems. Here, we consider two models for interacting disease-like dynamics with asymmetric interactions and different associated time scales. Using rate equations for homogeneously mixed populations, we show that the stationary prevalences and phase diagrams of each model behave differently with respect to variations of the relative time scales. We also characterize in detail the regime where transien…
The interpretation of sampling data plays a crucial role in policy response to the spread of a disease during an epidemic, such as the COVID-19 epidemic of 2020. However, this is a non-trivial endeavor due to the complexity of real world conditions and limits to the availability of diagnostic tests, which necessitate a bias in testing favoring symptomatic individuals. A thorough understanding of sampling confidence and bias is necessary in order make accurate conclusions. In this manuscript, we provide a stochastic model of sampling for assessing confidence in disease metrics such as trend det…
Background: Animal trade plays an important role for the spread of infectious diseases in livestock populations. As a case study, we consider pig trade in Germany, where trade actors (agricultural premises) form a complex network. The central question is how infectious diseases can potentially spread within the system of trade contacts. We address this question by analyzing the underlying network of animal movements. Methodology/Findings: The considered pig trade dataset spans several years and is analyzed with respect to its potential to spread infectious diseases. Focusing on measurements of…
Dynamic properties of spreading infection through a heterogeneous population are studied numerically and analytically using a dynamic variant of Watts and Strogatz Small World Network-based stochastic Susceptible-Exposed-Infectious-Removed (SEIR) epidemic model. This model includes the main realistic parameters usually characterizing transmissible diseases, such as the force of infection, latency and infection times. As far as the latency time remains smaller than that of infection, the proportion of infected individuals increases exponentially with time, otherwise an oscillatory behavior appe…
Several European countries have suspended the inoculation of the AstraZeneca vaccine out of suspicion of causing deep vein thrombosis. In this letter we report some Fermi estimates performed using a stochastic model aimed at making a risk-benefit analysis of the interruption of the delivery of the AstraZeneca vaccine in France and Italy. Our results clearly show that excess deaths due to the interruption of the vaccination campaign injections largely overrun those due to thrombosis even in worst case scenarios of frequency and gravity of the vaccine side effects.
In a metapopulation network, infectious diseases spread widely because of the travel of individuals. In the present study, we consider a modified metapopulation Susceptible-Infected-Removed (SIR) model with a latent period, which we call the SHIR model. In the SHIR model, an infectious period is divided into two stages. In the first stage, which corresponds to the latent period, infectious individuals can travel. However, in the second stage, the same individuals cannot travel since they are seriously ill. Final size distributions of the metapopulation SIR and SHIR models are simulated with tw…
Epidemic spread in a population is traditionally modeled via compartmentalized models which represent the free evolution of disease in absence of any intervention policies. In addition, these models assume full observability of disease cases and do not account for under-reporting. We present a mathematical model, namely PolSIRD, which accounts for the under-reporting by introducing an observation mechanism. It also captures the effects of intervention policies on the disease spread parameters by leveraging intervention policy data along with the reported disease cases. Furthermore, we allow ou…
To contain the propagation of emerging diseases that are transmissible from human to human, non-pharmaceutical interventions (NPIs) aimed at reducing the interactions between humans are usually implemented. One example of the latter kind of measures is social distancing, which can be either policy-driven or can arise endogenously in the population as a consequence of the fear of infection. However, if NPIs are lifted before the population reaches herd immunity, further re-introductions of the pathogen would lead to secondary infections. Here we study the effects of different social distancing …
Since 1927, until recently, models describing the spread of disease have mostly been of the SIR-compartmental type, based on the assumption that populations are homogeneous and well-mixed. The focus of these models have typically been on large-scale analysis of scenarios such as cities, nations or even world scale. SIR models are appealing because of their simplicity, but their parameters, especially the transmission rate, are complex and depend on a number of factors, which makes it hard to predict how a change of a single environmental, demographic, or epidemiological factor will affect the …
Disease, opinions, ideas, gossip, etc. all spread on social networks. How these networks are connected (the network structure) influences the dynamics of the spreading processes. By investigating these relationships one gains understanding both of the spreading itself and the structure and function of the contact network. In this chapter, we will summarize the recent literature using simulation of spreading processes on top of empirical contact data. We will mostly focus on disease simulations on temporal proximity networks -- networks recording who is close to whom, at what time -- but also c…
Social network analysis is now widely used to investigate the dynamics of infectious disease spread from person to person. Vaccination dramatically disrupts the disease transmission process on a contact network, and indeed, sufficiently high vaccination rates can disrupt the process to such an extent that disease transmission on the network is effectively halted. Here, we build on mounting evidence that health behaviors - such as vaccination, and refusal thereof - can spread through social networks through a process of complex contagion that requires social reinforcement. Using network simulat…
The interplay of biological, social, structural and random factors makes disease forecasting extraordinarily complex. The course of an epidemic exhibits average growth dynamics determined by features of the pathogen and the population, yet also features significant variability reflecting the stochastic nature of disease spread. In this work, we reframe a stochastic branching process analysis in terms of probability generating functions and compare it to continuous time epidemic simulations on networks. In doing so, we predict the diversity of emerging epidemic courses on both homogeneous and h…
COVID-19 is a new type of coronavirus disease which is caused by severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2). It originated in China in the month of December 2019 and quickly started to spread within the country. On 31st December 2019, it was first reported to country office of World Health Organization (WHO) in China. Since then, it has spread to most of the countries around the globe. However, there has been a recent rise in trend in believing that it would go away during summer days, which has not yet been properly investigated. In this paper, relationship of daily number o…
Non-pharmaceutical measures such as social distancing, can play an important role to control an epidemic in the absence of vaccinations. In this paper, we study the impact of social distancing on epidemics for which it is executable. We use a mathematical model combining human mobility and disease spreading. For the mobility dynamics, we design an agent based model consisting of pedestrian dynamics with a novel type of force to resemble social distancing in crowded sites. For the spreading dynamics, we consider the compartmental SIE dynamics plus an indirect transmission with the footprints of…
In this article, we discuss an age-structured SIR model in which disease not only spread through direct person to person contacts for e.g. infection due to surface contamination but it can also spread through indirect contacts. It is evident that age also plays a crucial role in SARS virus infection including COVID-19 infection. We formulate our model as an abstract semilinear Cauchy problem in an appropriate Banach space to show the existence of solution and also show the existence of steady states. It is assumed in this work that the population is in a demographic stationary state and show t…
The symptoms of many infectious diseases influence their host to withdraw from social activity limiting their own potential to spread. Successful transmission therefore requires the onset of infectiousness to coincide with a time when its host is socially active. Since social activity and infectiousness are both temporal phenomena, we hypothesize that diseases are most pervasive when these two processes are synchronized. We consider disease dynamics that incorporate a behavioral response that effectively shortens the infectious period of the disease. We apply this model to data collected from …
Understanding the dynamics of infectious disease spread in a heterogeneous population is an important factor in designing control strategies. Here, we develop a novel tensor-driven multi-compartment version of the classic Susceptible-Infected-Recovered (SIR) model and apply it to Internet data to reveal information about the complex spatial structure of disease spread. The model is used to analyze state-level Google search data from the US pertaining to two viruses, Respiratory Syncytial Virus (RSV), and West Nile Virus (WNV). We fit the data with correlations of $R^2=0.70$, and $0.52$ for RSV…
We combine a pedestrian dynamics model with a contact tracing method to simulate the initial spreading of a highly infectious airborne disease in a confined environment. We focus on a medium size population (up to 1000 people) with a small number of infectious people (1 or 2) and the rest of the people is divided between immune and susceptible. We adopt a space-continuous model that represents pedestrian dynamics by the forces acting on them, i.e. a microscopic force-based model. Once discretized, the model results in a high-dimensional system of second order ordinary differential equations. B…
We propose a mathematical model to analyze the time evolution of the total number of infected population with Covid-19 disease at a region in the ongoing pandemic. Using the available data of Covid-19 infected population on various countries we formulate a model which can successfully track the time evolution from early days to the saturation period in a given wave of this infectious disease. It involves a set of effective parameters which can be extracted from the available data. Using those parameters the future trajectories of the disease spread can also be projected. A set of differential …
Outbreaks of infectious diseases present a global threat to human health and are considered a major health-care challenge. One major driver for the rapid spatial spread of diseases is human mobility. In particular, the travel patterns of individuals determine their spreading potential to a great extent. These travel behaviors can be captured and modelled using novel location-based data sources, e.g., smart travel cards, social media, etc. Previous studies have shown that individuals who cannot be characterized by their most frequently visited locations spread diseases farther and faster; howev…
By the end of July 2020, the COVID-19 pandemic had infected more than seventeen million people and had spread to almost all countries worldwide. In response, many countries all over the world have used different methods to reduce the infection rate, such as including case isolation, the closure of schools and universities, banning public events, and mostly forcing social distancing, including local and national lockdowns. We use a Monte-Carlo (MC) based algorithm to predict the virus infection rate for different population densities using the most recent epidemic data in our work. We test the …
This study presents a method to discover an outbreak of an infectious disease in a region for which data are missing, but which is at work as a disease spreader. Node discovery for the spread of an infectious disease is defined as discriminating between the nodes which are neighboring to a missing disease spreader node, and the rest, given a dataset on the number of cases. The spread is described by stochastic differential equations. A perturbation theory quantifies the impact of the missing spreader on the moments of the number of cases. Statistical discriminators examine the mid-body or tail…
A novel predictive modeling framework for the spread of infectious diseases using high dimensional partial differential equations is developed and implemented. A scalar function representing the infected population is defined on a high-dimensional space and its evolution over all directions is described by a population balance equation (PBE). New infections are introduced among the susceptible population from non-quarantined infected population based on their interaction, adherence to distancing norms, hygiene levels and any other societal interventions. Moreover, recovery, death, immunity and…
Metapopulation models describing cities with different populations coupled by the travel of individuals are of great importance in the understanding of disease spread on a large scale. An important example is the Rvachev-Longini model [{\it Math. Biosci.} {\bf 75}, 3-22 (1985)] which is widely used in computational epidemiology. Few analytical results are however available and in particular little is known about paths followed by epidemics and disease arrival times. We study the arrival time of a disease in a city as a function of the starting seed of the epidemics. We propose an analytical An…
In this paper we study the household-structure SIS epidemic spreading on general complex networks. The household structure gives us the way to distinguish inner and the outer infection rate. Unlike household-structure models on homogenous networks, such as regular and random networks, here we consider heterogeneous networks with arbitrary degree distribution p(k). First we introduce the epidemic model. Then rate equations under mean field appropriation and computer simulations are used here to analyze our model. Some unique phenomena only existing in divergent network with household structure …
There is a rich history of models for the interaction of a biological contagion like influenza with the spread of related information such as an influenza vaccination campaign. Recent work on the spread of interacting contagions on networks has highlighted that these interacting contagions can have counter-intuitive interplay with network structure. Here we generalize one of these frameworks to tackle three important features of the spread of awareness and disease: one, we model the dynamics on highly clustered, cliquish, networks to mimic the role of workplaces and households; two, the awaren…
Models of disease spreading are critical for predicting infection growth in a population and evaluating public health policies. However, standard models typically represent the dynamics of disease transmission between individuals using macroscopic parameters that do not accurately represent person-to-person variability. To address this issue, we present a dynamic network model that provides a straightforward way to incorporate both disease transmission dynamics at the individual scale as well as the full spatiotemporal history of infection at the population scale. We find that disease spreads …
Two crucial elements facilitate the understanding and control of communicable disease spread within a social setting. These components are, the underlying contact structure among individuals that determines the pattern of disease transmission; and the evolution of this pattern over time. Mathematical models of infectious diseases, which are in principle analytically tractable, use two general approaches to incorporate these elements. The first approach, generally known as compartmental modeling, addresses the time evolution of disease spread at the expense of simplifying the pattern of transmi…
Diffusion processes are governed by external triggers and internal dynamics in complex systems. Timely and cost-effective control of infectious disease spread critically relies on uncovering the underlying diffusion mechanisms, which is challenging due to invisible causality between events and their time-evolving intensity. We infer causal relationships between infections and quantify the reflexivity of a meta-population, the level of feedback on event occurrences by its internal dynamics (likelihood of a regional outbreak triggered by previous cases). These are enabled by our new proposed mod…
Land conversion and the resulting contact between domesticated and wild species has arguably been the single largest contributor to the emergence of novel epizootic and zoonotic diseases in the past century. An unintended consequence of these interactions is zoonotic or epizootic disease spillovers from wild species to humans and their domesticates. Disease spillovers are edge effects of land conversion and are sensitive to the size and shape of converted areas. We combine spatial metrics from landscape ecology with theoretical epidemiological models to understand how the size and shape of lan…
In this paper, we introduce a modified epidemic model on regular and scale-free networks respectively. We consider the birth rate $δ$, cure rate $γ$, infection rate $λ$, $α$ from the infectious disease, and death rate $β$ from other factors. Through mean-field analysis, we find that on regular network there is an epidemic threshold $λ_{c}$ dependent on the parameters $δ$, $γ$, $α$, and $β$; while for power law degree distribution network epidemic threshold is absent in the thermodynamic limit. The result is the same as that of the standard SIS model. This reminds us the structure of the networ…
We consider the edge-based compartmental models for infectious disease spread introduced in Part I. These models allow us to consider standard SIR diseases spreading in random populations. In this paper we show how to handle deviations of the disease or population from the simplistic assumptions of Part I. We allow the population to have structure due to effects such as demographic detail or multiple types of risk behavior the disease to have more complicated natural history. We introduce these modifications in the static network context, though it is straightforward to incorporate them into d…
The recent outbreak of a novel coronavirus and its rapid spread underlines the importance of understanding human mobility. Enclosed spaces, such as public transport vehicles (e.g. buses and trains), offer a suitable environment for infections to spread widely and quickly. Investigating the movement patterns and the physical encounters of individuals on public transit systems is thus critical to understand the drivers of infectious disease outbreaks. For instance previous work has explored the impact of recurring patterns inherent in human mobility on disease spread, but has not considered othe…
The emergence of diseases such as Zika and Ebola has highlighted the need to understand the role of sexual transmission in the spread of diseases with a primarily non-sexual transmission route. In this paper we develop a number of low-dimensional models which are appropriate for a range of assumptions for how a disease will spread if it has sexual transmission through a sexual contact network combined with some other transmission mechanism, such as direct contact or vectors. The equations derived provide exact predictions for the dynamics of the corresponding simulations in the large populatio…
We study metapopulation models for the spread of epidemics in which different subpopulations (cities) are connected by fluxes of individuals (travelers). This framework allows to describe the spread of a disease on a large scale and we focus here on the computation of the arrival time of a disease as a function of the properties of the seed of the epidemics and of the characteristics of the network connecting the various subpopulations. Using analytical and numerical arguments, we introduce an easily computable quantity which approximates this average arrival time. We show on the example of a …
Routine public health surveillance of notifiable infectious diseases gives rise to weekly counts of reported cases -- possibly stratified by region and/or age group. We investigate how an age-structured social contact matrix can be incorporated into a spatio-temporal endemic-epidemic model for infectious disease counts. To illustrate the approach, we analyze the spread of norovirus gastroenteritis over 6 age groups within the 12 districts of Berlin, 2011-2015, using contact data from the POLYMOD study. The proposed age-structured model outperforms alternative scenarios with homogeneous or no m…
The study of infectious disease has been of interest to ecologists since long. The initiation of epidemic and the long term disease dynamics are largely influenced by the nature of the underlying consumer (host)-resource dynamics. Ecological traits of such systems may be often modulated by toxins released in the environment due to ongoing anthropogenic activities. This, in addition to toxin-mediated alteration of epidemiological traits, has a significant impact on disease progression in ecosystems which is quite less studied. In order to address this, we consider a mathematical model of diseas…
We consider a model for an epidemic in a population that occupies geographically distinct locations. The disease is spread within subpopulations by contacts between infective and susceptible individuals, and is spread between subpopulations by the migration of infected individuals. We show how susceptible individuals can act collectively to limit the spread of disease during the initial phase of an epidemic, by specifying the distribution that minimises the growth rate of the epidemic when the infectives are migrating so as to maximise the growth rate. We also give an explicit strategy that mi…
Deterministic models are developed for the spatial spread of epidemic diseases in geographical settings. The models are focused on outbreaks that arise from a small number of infected hosts imported into sub-regions of the geographical settings. The goal is to understand how spatial heterogeneity influences the transmission dynamics of the susceptible and infected populations. The models consist of systems of partial differential equations with diffusion terms describing the spatial spread of the underlying microbial infectious agents. The model is compared with real data from seasonal influen…
We develop simple models for the global spread of infectious diseases, emphasizing human mobility via air travel and the variation of public health infrastructure from region to region. We derive formulas relating the total and peak number of infections in two countries to the rate of travel between them and their respective epidemiological parameters.
During the COVID-19 pandemic, conflicting opinions on physical distancing swept across social media, affecting both human behavior and the spread of COVID-19. Inspired by such phenomena, we construct a two-layer multiplex network for the coupled spread of a disease and conflicting opinions. We model each process as a contagion. On one layer, we consider the concurrent evolution of two opinions -- pro-physical-distancing and anti-physical-distancing -- that compete with each other and have mutual immunity to each other. The disease evolves on the other layer, and individuals are less likely (re…
