In this work, we review the figures used to characterize an epidemic outbreak most. Particular attention is drawn to epidemic spreading at time-varying transition rates. A time-varying SIR-like model is used to describe the epidemic dynamics, from which closed equations relating parameters and key quantities like reproduction ratio and doubling time are derived. The definition and computation of these metrics are revisited in the context of the general solution to the time-varying model dynamics, focusing on the similarities and differences with the time-invariant case. Further, the prediction of these metrics, that is of the disease evolution, as response to different scenarios is also investigated.
The 2014 epidemic of the Ebola virus is governed by a genetically diverse viral population. In the early Sierra Leone outbreak, a recent study has identified new mutations that generate genetically distinct sequence clades. Here we find evidence that major Sierra Leone clades have systematic differences in growth rate and reproduction number. If this growth heterogeneity remains stable, it will generate major shifts in clade frequencies and influence the overall epidemic dynamics on time scales within the current outbreak. Our method is based on simple summary statistics of clade growth, which…
During outbreaks of serious infectious diseases many individuals closely follow media reports of the outbreak and take steps, including self-isolation, to protect themselves from infection and possibly death. Self-isolation can take many forms including restricting local and long-distance travel, using face masks, and choosing to receive a vaccine. We use mathematical modeling to show that public health agencies working together with the media can significantly decrease the severity of an outbreak by providing timely and accurate accounts of the numbers of new infections and deaths. Our model …
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…
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…
We analyze the time-series data for the onset of A(H1N1)v influenza pandemic in India during the period June 1- September 30, 2009. Using a variety of statistical fitting procedures, we obtain a robust estimate of the exponential growth rate $\langle λ\rangle \simeq 0.15$. This corresponds to a basic reproductive number $R_0 \simeq 1.45$ for influenza A(H1N1)v in India, a value which lies towards the lower end of the range of values reported for different countries affected by the pandemic.
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…
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.
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…
A better characterization of the early growth dynamics of an epidemic is needed to dissect the important drivers of disease transmission. We introduce a 2-parameter generalized-growth model to characterize the ascending phase of an outbreak and capture epidemic profiles ranging from sub-exponential to exponential growth. We test the model against empirical outbreak data representing a variety of viral pathogens and provide simulations highlighting the importance of sub-exponential growth for forecasting purposes. We applied the generalized-growth model to 20 infectious disease outbreaks repres…
Global strategies to contain a pandemic, such as social distancing and protective measures, are designed to reduce the overall transmission rate between individuals. Despite such measures, essential institutions, including hospitals, schools, and food producing plants, remain focal points of local outbreaks. Here we develop a model for the stochastic outbreak dynamics in such local communities. We derive analytical expressions for the probability of containment of the outbreak, which is complementary to the probability of seeding a deterministically growing epidemic. This probability depends o…
Despite the enormous relevance of zoonotic infections to world- wide public health, and despite much effort in modeling individual zoonoses, a fundamental understanding of the disease dynamics and the nature of outbreaks arising in such systems is still lacking. We introduce a simple stochastic model of susceptible-infected- recovered dynamics in a coupled animal-human metapopulation, and solve analytically for several important properties of the cou- pled outbreaks. At early timescales, we solve for the probability and time of spillover, and the disease prevalence in the animal population at …
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 …
The use of model experimental systems and mathematical models is important to further understanding of infectious disease dynamics and strategize disease mitigation. Gyrodactylids are helminth ectoparasites of teleost fish which have many dynamical characteristics of microparasites but offer the advantage that they can be quantified and tracked over time, allowing further insight into within-host and epidemic dynamics. In this paper, we design a model to describe host-parasite dynamics of the well-studied guppy-Gyrodactylus turnbulli system, using experimental data to estimate parameters and v…
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 …
Biocomplexity Institute, University of Virginia（生物复杂性研究所，弗吉尼亚大学）；Johns Hopkins University（约翰霍普金斯大学）；Biocomplexity Institute and Department of Computer Science, University of Virginia（生物复杂性研究所与计算机科学系，弗吉尼亚大学）；University of Pittsburgh（匹兹堡大学）
The Richards family models comprise a well-known set of models with useful parameters to describe several aspects of disease outbreaks. Some of these models have been used to study the current Covid-19 pandemic. However, there seems to be confusion regarding the discrimination among competing models. In this paper a detailed application of Akaikes information approach is used to discern among models using data from The European Union, The United States and The United Kingdom. We argue that the epidemiological characteristics derived from competing models should be examined to complement the se…
Epidemiological models may give some basic guidelines for public health practitioners, allowing to analyze issues that can influence the strategies to prevent and fight a disease. To be used in decision-making, however, a mathematical model must be carefully parameterized and validated with epidemiological and entomological data. Here a SIR (S for susceptible, I for infectious, R for recovered individuals) and ASI (A for the aquatic phase of the mosquito, S for susceptible and I for infectious mosquitoes) epidemiological model describing a dengue disease is presented, as well as the associated…
High-throughput molecular screening is required to investigate the origin and diffusion of antimicrobial resistance in pathogen outbreaks. The most frequent cause of human infection is Escherichia coli, which is dominated by sequence type 131 (ST131), a set of rapidly radiating pandemic clones. The highly infectious clades of ST131 originated firstly by a mutation enhancing virulence and adhesion. Secondly, single-nucleotide polymorphisms occurred enabling fluoroquinolone-resistance, which is near-fixed in all ST131. Thirdly, broader resistance through beta-lactamases has been gained and lost …
In this paper we model the spreading of the SARS-CoV-2 in Mexico by introducing a new stochastic approximation constructed from first principles, structured on the basis of a Latent-Infectious- (Recovered or Deceased) (LI(RD)) compartmental approximation, where the number of new infected individuals caused by a single infectious individual per unit time (a day), is a random variable of a Poisson distribution and whose parameter is modulated through a weight-like time-dependent function. The weight function serves to introduce a time dependence to the average number of new infections and as we …
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…
We study the effects of switching social contacts as a strategy to control epidemic outbreaks. Connections between susceptible and infective individuals can be broken by either individual, and then reconnected to a randomly chosen member of the population. It is assumed that the reconnecting individual has no previous information on the epidemiological condition of the new contact. We show that reconnection can completely suppress the disease, both by continuous and discontinuous transitions between the endemic and the infection-free states. For diseases with an asymptomatic phase, we analyze …
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.
In this paper we study the susceptible-infectious (SI) epidemiological model using dynamical graphs. Dynamical structures have been recently applied in many areas including complex systems. Dynamical structures include the mutual interaction between the structure topology and the characteristics of its members. Dynamical graphs applied to epidemics consider generally that the nodes are individuals and the links represent different classes of relationships between individuals with the potential to transmit the disease. The main aim in this article is to study the evolution of the SI epidemiolog…
The SARS-CoV-2 infectious outbreak has rapidly spread across the globe and precipitated varying policies to effectuate physical distancing to ameliorate its impact. In this study, we propose a new hybrid machine learning model, SIRNet, for forecasting the spread of the COVID-19 pandemic that couples with the epidemiological models. We use categorized spatiotemporally explicit cellphone mobility data as surrogate markers for physical distancing, along with population weighted density and other local data points. We demonstrate at varying geographical granularity that the spectrum of physical di…
We present a computational modeling framework for data-driven simulations and analysis of infectious disease spread in large populations. For the purpose of efficient simulations, we devise a parallel solution algorithm targeting multi-socket shared memory architectures. The model integrates infectious dynamics as continuous-time Markov chains and available data such as animal movements or aging are incorporated as externally defined events. To bring out parallelism and accelerate the computations, we decompose the spatial domain and optimize cross-boundary communication using dependency-aware…
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.
Though pandemic preparedness has been a focus of public health planning for centuries, during which our understanding of infectious disease dynamics has grown, our methodologies for managing outbreaks have remained relatively unchanged. We propose leveraging this history to identify opportunities for actual progress. We contrast current plans with historical outbreak control measures and isolate how the complexities of a modern era yield additional challenges in how best to anticipate and mitigate outbreaks. We analyze a diversity of publicly available modern preparedness plans against the con…
I employ a simple mathematical model of an epidemic process to evaluate how four basic quantities: the reproduction number (R), the numbers of sensitive (S) and infectious individuals(I), and total community size (N) affect strategies to control COVID-19. Numerical simulations show that strict suppression measures at the beginning of an epidemic can create low infectious numbers, which thereafter can be managed by mitigation measures over longer periods to flatten the epidemic curve. The stronger the suppression measure, the faster it achieves the low numbers of infections that are conducive t…
When controlling an emerging outbreak of an infectious disease it is essential to know the key epidemiological parameters, such as the basic reproduction number $R_0$ and the control effort required to prevent a large outbreak. These parameters are estimated from the observed incidence of new cases and information about the infectious contact structures of the population in which the disease spreads. However, the relevant infectious contact structures for new, emerging infections are often unknown or hard to obtain. Here we show that for many common true underlying heterogeneous contact struct…
We define data analyses to monitor a change in R, the average number of secondary cases caused by a typical infected individual. The input dataset consists of incident cases partitioned into outbreaks, each initiated from a single index case. We split of the input dataset into two successive subsets, to evaluate two successive R values, according to the Bayesian paradigm. We used the Bayes factor between the model with two different R values and that with a single R value to justify that the change in R is statistically significant. We validated our approach using simulated data, generated usi…
Once an epidemic outbreak has been effectively contained through non-pharmaceutical interventions, a safe protocol is required for the subsequent release of social distancing restrictions to prevent a disastrous resurgence of the infection. We report individual-based numerical simulations of stochastic susceptible-infectious-recovered model variants on four distinct spatially organized lattice and network architectures wherein contact and mobility constraints are implemented. We robustly find that the intensity and spatial spread of the epidemic recurrence wave can be limited to a manageable e…
For a recently derived pairwise model of network epidemics with non-Markovian recovery, we prove that under some mild technical conditions on the distribution of the infectious periods, smaller variance in the recovery time leads to higher reproduction number, and consequently to a larger epidemic outbreak, when the mean infectious period is fixed. We discuss how this result is related to various stochastic orderings of the distributions of infectious periods. The results are illustrated by a number of explicit stochastic simulations, suggesting that their validity goes beyond regular networks…
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…
Contact patterns in populations fundamentally influence the spread of infectious diseases. Current mathematical methods for epidemiological forecasting on networks largely assume that contacts between individuals are fixed, at least for the duration of an outbreak. In reality, contact patterns may be quite fluid, with individuals frequently making and breaking social or sexual relationships. Here we develop a mathematical approach to predicting disease transmission on dynamic networks in which each individual has a characteristic behavior (typical contact number), but the identities of their c…
Branching processes are widely used to model evolutionary and population dynamics as well as the spread of infectious diseases. To characterize the dynamics of their growth or spread, the basic reproduction number $R_0$ has received considerable attention. In the context of infectious diseases, it is usually defined as the expected number of secondary cases produced by an infectious case in a completely susceptible population. Typically $R_0>1$ indicates that an outbreak is expected to continue and to grow exponentially, while $R_0<1$ usually indicates that an outbreak is expected to terminate…
Since the end of 2019 an outbreak of a new strain of coronavirus, called 2019--nCoV, is reported from China and later other parts of the world. Since January 21, WHO reports daily data on confirmed cases and deaths from both China and other countries. In this work we present some discrete and continuous models to discribe the disease dynamics in China and estimate the needed epidemiological parameters. Good agreement with the current dynamics has be found for both a discrete transmission model and a slightly modified SIR-model.
The spread of infectious diseases near the epidemic threshold is investigated. Scaling laws for the size and the duration of outbreaks originating from a single infected individual in a large susceptible population are obtained. The maximal size of an outbreak n_* scales as N^{2/3} with N the population size. This scaling law implies that the average outbreak size <n> scales as N^{1/3}. Moreover, the maximal and the average duration of an outbreak grow as t_* ~ N^{1/3} and <t> ~ ln N, respectively.
Sensitivity Analysis (SA) is a useful tool to measure the impact of changes in model parameters on the infection dynamics, particularly to quantify the expected efficacy of disease control strategies. SA has only been applied to epidemic models at the population level, ignoring the effect of within-host virus-with-immune-system interactions on the disease spread. Connecting the scales from individual to population can help inform drug and vaccine development. Thus the value of understanding the impact of immunological parameters on epidemiological quantities. Here we consider an age-since-infe…
Probability generating functions for epidemiological time series
The Roux Institute, Northeastern University（东北大学鲁克斯学院）；Network Science Institute, Northeastern University（东北大学网络科学研究所）；School of Electrical Engineering and Computer Science, University of Ottawa（渥太华大学电气与计算机工程学院）
