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.
In two previous papers, I introduced SuperSpreader (SS) epidemic models, offered some theoretical discussion of prevention issues, and fitted some models to data derived from published accounts of the ongoing MERS epidemic (concluding that a pandemic is likely). Continuing on this theme, here I discuss prediction: whether, in a disease outbreak driven by superspreader events, a rigorous decision point---meaning a declaration that a pandemic is imminent---can be defined. I show that all sources of prediction bias contribute to generating false negatives (i.e., discounting the chance of a pandem…
The annual occurrence of many infectious diseases remains a constant burden to public health systems. The seasonal patterns in respiratory disease incidence observed in temperate regions have been attributed to the impact of environmental conditions on pathogen survival. A model describing the transmission of an infectious disease by means of a pathogenic state capable of surviving in an environmental reservoir outside of its host organism is presented in this paper. The ratio of pathogen lifespan to the duration of the infectious disease state is found to be a critical parameter in determinin…
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.
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…
Interaction patterns among individuals play vital roles in spreading infectious diseases. Understanding these patterns and integrating their impact in modeling diffusion dynamics of infectious diseases are important for epidemiological studies. Current network-based diffusion models assume that diseases transmit through interactions where both infected and susceptible individuals are co-located at the same time. However, there are several infectious diseases that can transmit when a susceptible individual visits a location after an infected individual has left. Recently, we introduced a diffus…
The transmission dynamics of some infectious diseases is related to the contact structure between individuals in a network. We used five algorithms to generate contact networks with different topological structure but with the same scale-free degree distribution. We simulated the spread of acute and chronic infectious diseases on these networks, using SI (Susceptible - Infected) and SIS (Susceptible - Infected - Susceptible) epidemic models. In the simulations, our objective was to observe the effects of the topological structure of the networks on the dynamics and prevalence of the simulated …
We introduce a Python package for modeling and studying the spread of infectious diseases using an agent-based SEIR style epidemiological model with a focus on university campuses. This document explains the epidemiological model used in the package and gives examples highlighting the ways that the package can be used.
Urban areas with larger and more connected populations offer an auspicious environment for contagion processes such as the spread of pathogens. Empirical evidence reveals a systematic increase in the rates of certain sexually transmitted diseases (STDs) with larger urban population size. However, the main drivers of these systemic infection patterns are still not well understood, and rampant urbanization rates worldwide makes it critical to advance our understanding on this front. Using confirmed-cases data for three STDs in US metropolitan areas, we investigate the scaling patterns of infecti…
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…
Spatial big data have the "velocity," "volume," and "variety" of big data sources and additional geographic information about the record. Digital data sources, such as medical claims, mobile phone call data records, and geo-tagged tweets, have entered infectious disease epidemiology as novel sources of data to complement traditional infectious disease surveillance. In this work, we provide examples of how spatial big data have been used thus far in epidemiological analyses and describe opportunities for these sources to improve public health coordination and disease mitigation strategies. In a…
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…
We propose a risk measure and construct an infectious diseases hazard map for India. Given an outbreak location, a hazard index is assigned to each city using an effective distance that depends on inter-city mobilities instead of geographical distance. We demonstrate its utility using an SIR model augmented with air, rail, and road data between top 446 cities. Simulations show that the effective distance from outbreak location reliably predicts the time of arrival of infection in other cities. The hazard index predictions compare well with the observed spread of SARS-CoV-2. The hazard map can …
Infectious disease outbreaks recapitulate biology: they emerge from the multi-level interaction of hosts, pathogens, and their shared environment. As a result, predicting when, where, and how far diseases will spread requires a complex systems approach to modeling. Recent studies have demonstrated that predicting different components of outbreaks--e.g., the expected number of cases, pace and tempo of cases needing treatment, demand for prophylactic equipment, importation probability etc.--is feasible. Therefore, advancing both the science and practice of disease forecasting now requires testin…
In this work we describe a non-parametric disease model that links the temporal change of the prevalence of an infectious disease to the incidence and the recovery rates. The model is only based on the common epidemiological measures incidence and recovery rate. As an application, the model is used to calculate the prevalence of influenza in Germany for a hypothetical birth cohort during 2001 and 2013.
Disease outbreaks, such as those of Severe Acute Respiratory Syndrome in 2003 and the 2009 pandemic A(H1N1) influenza, have highlighted the potential for airborne transmission in indoor environments. Respirable pathogen-carrying droplets provide a vector for the spatial spread of infection with droplet transport determined by diffusive and convective processes. An epidemiological model describing the spatial dynamics of disease transmission is presented. The effects of an ambient airflow, as an infection control, are incorporated leading to a delay equation, with droplet density dependent on t…
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…
Infectious disease models can be of great use for understanding the underlying mechanisms that influence the spread of diseases and predicting future disease progression. Modeling has been increasingly used to evaluate the potential impact of different control measures and to guide public health policy decisions. In recent years, there has been rapid progress in developing spatio-temporal modeling of infectious diseases and an example of such recent developments is the discrete time individual-level models (ILMs). These models are well developed and provide a common framework for modeling many…
We consider multiple diseases spreading in a static Configuration Model network. We make standard assumptions that infection transmits from neighbor to neighbor at a disease-specific rate and infected individuals recover at a disease-specific rate. Infection by one disease confers immediate and permanent immunity to infection by any disease. Under these assumptions, we find a simple, low-dimensional ordinary differential equations model which captures the global dynamics of the infection. The dynamics depend strongly on initial conditions. Although we motivate this article with infectious dise…
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 …
We develop a theory of first passage processes in stochastic non-equilibrium systems of birth-death type using two closely related epidemiological models as examples. Our method employs the probability generating function technique in conjunction with the eikonal approximation. In this way the problem is reduced to finding the optimal path to extinction: a heteroclinic trajectory of an effective multi-dimensional classical Hamiltonian system. We compute this trajectory and mean extinction time of the disease numerically and uncover a non-monotone, spiral path to extinction of a disease. We als…
A model describing the dynamics related to the spreading of non-lethal infectious diseases in a fixed-size population is proposed. The model consists of a non-linear delay-differential equation describing the time evolution of the increment in the number of infectious individuals and depends upon a limited number of parameters. Predictions are in good qualitative agreement with data on influenza.
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 global public health landscape is perpetually challenged by the looming threat of infectious diseases. Central to addressing this concern is the imperative to prevent and manage disease transmission during pandemics, particularly in unique settings. This study addresses the transmission dynamics of infectious diseases within conference venues, presenting a computational model designed to simulate transmission processes within a condensed timeframe (one day), beginning with sporadic cases. Our model intricately captures the activities of individual attendees within the conference venue, enc…
During an infectious disease outbreak, biases in the data and complexities of the underlying dynamics pose significant challenges in mathematically modelling the outbreak and designing policy. Motivated by the ongoing response to COVID-19, we provide a toolkit of statistical and mathematical models beyond the simple SIR-type differential equation models for analysing the early stages of an outbreak and assessing interventions. In particular, we focus on parameter estimation in the presence of known biases in the data, and the effect of non-pharmaceutical interventions in enclosed subpopulation…
Runs of homozygosity (ROH) are contiguous stretches of homozygous genome which likely reflect transmission from common ances- tors and can be used to track the inheritance of haplotypes of interest. In the present paper, ROH were extracted from 50K SNPs and used to detect regions of the genome associated with susceptibility to diseases in a population of 468 Holstein-Frisian cows. Diagnosed diseases were categorised as infectious diseases, metabolic syndromes, mastitis, reproductive diseases and locomotive disorders. ROH associated with infectious diseases, mastitis and locomotive disorders we…
The dynamics of economies and infectious disease are inexorably linked: economic well-being influences health (sanitation, nutrition, treatment capacity, etc.) and health influences economic well-being (labor productivity lost to sickness and disease). Often societies are locked into "poverty traps" of poor health and poor economy. Here, using a simplified coupled disease-economic model with endogenous capital growth we demonstrate the formation of poverty traps, as well as ways to escape them. We suggest two possible mechanisms of escape both motivated by empirical data: one, through an influ…
Human mobility, contact patterns, and their interplay are key aspects of our social behavior that shape the spread of infectious diseases across different regions. In the light of new evidence and data sets about these two elements, epidemic models should be refined to incorporate both the heterogeneity of human contacts and the complexity of mobility patterns. Here, we propose a theoretical framework that allows accommodating these two aspects in the form of a set of Markovian equations. We validate these equations with extensive mechanistic simulations and derive analytically the epidemic th…
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.
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 thi…
In this paper we first introduce the general stochastic epidemic model for the spread of infectious diseases. Then we give methods for inferring model parameters such as the basic reproduction number $R_0$ and vaccination coverage $v_c$ assuming different types of data from an outbreak such as final outbreak details and temporal data or observations from an ongoing outbreak. Both individual heterogeneities and heterogeneous mixing are discussed. We also provide an overview of statistical methods to perform parameter estimation for stochastic epidemic models. In the last section we describe the…
Humans, domestic animals, orchard crops, and ornamental plants are commonly treated with antibiotics in response to bacterial infection. By curing infectious individuals, antibiotic therapy might limit the spread of contagious disease among hosts. But an antibiotic`s suppression of within-host pathogen density might also reduce the probability that the host is otherwise removed from infectious status prior to therapeutic recovery. When rates of both recovery via treatment and other removal events (e.g., isolation or mortality) depend directly on within-host pathogen density, antibiotic treatme…
With the covid-19 pandemic still ongoing and an enormous amount of test data available, the lessons learned over the last two years need to be developed to a point where they can provide understanding for tackling new variants and future diseases. The SIR-model commonly used to model disease spread, predicts exponential initial growth, which helps establish the infectiousness of a disease in the early days of an outbreak. Unfortunately, the exponential growth becomes muddied by spatial, finite-size, and non-equilibrium effects in realistic systems, and robust estimates that may be used in pred…
Humans, domestic animals, orchard crops, and ornamental plants are commonly treated with antibiotics in response to bacterial infection. By curing infectious individuals, antibiotic therapy might limit the spread of contagious disease among hosts. But antibiotic suppression of within-host pathogen density might also reduce the probability that the host is otherwise removed from infectious status before recovery. When rates of both recovery and removal (isolation or mortality) depend directly on within-host density, antibiotic therapy can relax the removal rate and so increase between-host dise…
In this paper, we consider the impact of time delay by media on the control of the disease. We set up a class of SISM epidemic model with the time delay and the cumulative density of awareness caused by media. The sufficient condition of global asymptotic stability of disease-free equilibrium is approved. We get the global stability of the epidemic equilibrium and the existence conditions of Hopf bifurcation. Numerical simulations are presented to illustrate the analytical results. Finally, we analyze the influence of parameters on the control of infectious disease by combining the data of H1N…
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…
Stochasticity and spatial heterogeneity are of great interest recently in studying the spread of an infectious disease. The presented method solves an inverse problem to discover the effectively decisive topology of a heterogeneous network and reveal the transmission parameters which govern the stochastic spreads over the network from a dataset on an infectious disease outbreak in the early growth phase. Populations in a combination of epidemiological compartment models and a meta-population network model are described by stochastic differential equations. Probability density functions are der…
We propose a mathematical model to analyze the effects of anti-infection behavior on the equilibrium states of an infectious disease. The anti-infection behavior is incorporated into a classical epidemiological SIR model, by considering the behavior adoption rate across the population as an additional variable. We consider also the effects on the adoption rate produced by the disease evolution, using a dynamic payoff function and an additional differential equation. The equilibrium states of the proposed model have remarkable characteristics: possible coexistence of two locally stable endemic …
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…
I study the spreading of infectious diseases on heterogeneous populations. I represent the population structure by a contact-graph where vertices represent agents and edges represent disease transmission channels among them. The population heterogeneity is taken into account by the agent's subdivision in types and the mixing matrix among them. I introduce a type-network representation for the mixing matrix allowing an intuitive understanding of the mixing patterns and the analytical calculations. Using an iterative approach I obtain recursive equations for the probability distribution of the o…
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…
Super-spreading events for infectious diseases occur when some infected individuals infect more than the average number of secondary cases. Several super-spreading individuals have been identified for the 2003 outbreak of severe acute respiratory syndrome (SARS). We develop a model for super-spreading events of infectious diseases, which is based on the outbreak of SARS. Using this model we describe two methods for estimating the parameters of the model, which we demonstrate with the small-scale SARS outbreak at the Amoy Gardens, Hong Kong, and the large-scale outbreak in the entire Hong Kong …
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:20…
We discuss the criticality in the stochastic SIR model for infectious diseases. We adopt the path-integral formalism for the propagation of infections among susceptible, infectious, and removed individuals, and perform the perturbative and nonperturbative analyses to evaluate the critical value of the basic reproduction number ${\cal R}$. In the perturbation theory, we calculate the mean values and the variances of the number of infectious individuals near the initial time, and find that the critical value ${\cal R}_{\text{c}}=1/3$-$2/3$ should be adopted in order to suppress the stochastic sp…
This paper analyses an SIRS-type model for infectious diseases with account for behavioural changes associated with the simultaneous spread of awareness in the population. Two types of awareness are included into the model: private awareness associated with direct contacts between unaware and aware populations, and public information campaign. Stability analysis of different steady states in the model provides information about potential spread of disease in a population, and well as about how the disease dynamics is affected by the two types of awareness. Numerical simulations are performed t…
Mathematical modeling is one of the key factors of the effective control of newly found infectious diseases, such as COVID-19. Our knowledge about the parameters and the course of the infection is highly limited in the beginning of the epidemic, hence computer implementation of the models have to be quick and flexible. The propensity matrix - update graph method we discuss in this paper serves as a convenient approach to efficiently implement age structured stochastic epidemic models. The code base we implemented for our forecasting work is also included in the attached GitHub repository.
It is well known that behavioral changes in contact patterns may significantly affect the spread of an epidemic outbreak. Here we focus on simple endemic models for recurrent epidemics, by modelling the social contact rate as a function of the available information on the present and past disease prevalence. We show that social behaviour change alone may trigger sustained oscillations. This indicates that human behavior might be a critical explaining factor of oscillations in time-series of endemic diseases. Finally, we briefly show how the inclusion of seasonal variations in contacts may impl…
A wide range of infectious diseases are both vertically and horizontally transmitted. Such diseases are spatially transmitted via multiple species in heterogeneous environments, typically described by complex meta-population models. The reproduction number is a critical metric predicting whether the disease can invade the meta-population system. This paper presents the reproduction number for a generic disease vertically and horizontally transmitted among multiple species in heterogeneous networks, where nodes are locations, and links reflect outgoing or incoming movement flows. The metapopula…
