A primary quantity of interest in the study of infectious diseases is the average number of new infections that an infected person produces. This so-called reproduction number has significant implications for the disease progression. There has been increasing literature suggesting that superspreading, the significant variability in number of new infections caused by individuals, plays an important role in the spread of SARS-CoV-2. In this paper, we consider the effect that such superspreading has on the estimation of the reproduction number and subsequent estimates of future cases. Accordingly, we employ a simple extension to models currently used in the literature to estimate the reproduction number and present a case-study of the progression of COVID-19 in Austria. Our models demonstrate that the estimation uncertainty of the reproduction number increases with superspreading and that this improves the performance of prediction intervals. Of independent interest is the derivation of a transparent formula that connects the extent of superspreading to the width of credible intervals for the reproduction number. This serves as a valuable heuristic for understanding the uncertainty surrounding diseases with superspreading.
The expected number of secondary infections arising from each index case, referred to as the reproduction or $R$ number, is a vital summary statistic for understanding and managing epidemic diseases. There are many methods for estimating $R$; however, few explicitly model heterogeneous disease reproduction, which gives rise to superspreading within the population. We propose a parsimonious discrete-time branching process model for epidemic curves that incorporates heterogeneous individual reproduction numbers. Our Bayesian approach to inference illustrates that this heterogeneity results in le…
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 be…
Epidemic disease spreading is conventionally often modelled and analyzed by means of rate and diffusion equations, following the paradigms of well-controlled chemical reactions and diffusive dynamics in a test tube. Yet, serious worries that this suggestive and appealing similarity might be a false friend were already voiced by the pioneers of mathematical epidemiology. A century later, we can draw on cross-fertilizations from network and game theory and the emerging field of eco-evolutionary dynamics to substantiate them. Epidemiological spreading is thereby revealed as a fundamentally hetero…
The effective reproduction number $R_t$ measures an infectious disease's transmissibility as the number of secondary infections in one reproduction time in a population having both susceptible and non-susceptible hosts. Current approaches do not quantify the uncertainty correctly in estimating $R_t$, as expected by the observed variability in contagion patterns. We elaborate on the Bayesian estimation of $R_t$ by improving on the Poisson sampling model of Cori et al. (2013). By adding an autoregressive latent process, we build a Dynamic Linear Model on the log of observed $R_t$s, resulting in …
IIT Bombay（印度理工学院孟买分校）；Max Planck Institute for Software Systems（马克斯·普朗克软件系统研究所）；ETH Zurich（苏黎世联邦理工学院）
Testing is recommended for all close contacts of confirmed COVID-19 patients. However, existing group testing methods are oblivious to the circumstances of contagion provided by contact tracing. Here, we build upon a well-known semi-adaptive pool testing method, Dorfman's method with imperfect tests, and derive a simple group testing method based on dynamic programming that is specifically designed to use the information provided by contact tracing. Experiments using a variety of reproduction numbers and dispersion levels, including those estimated in the context of the COVID-19 pandemic, show…
John A. Paulson School of Engineering and Applied Sciences, Harvard University（哈佛大学约翰·A·保尔森工程与应用科学学院）；T.H. Chan School of Public Health, Harvard University（哈佛大学陈曾熙公共卫生学院）
The COVID-19 pandemic provides new motivation for a classic problem in epidemiology: estimating the empirical rate of transmission during an outbreak (formally, the time-varying reproduction number) from case counts. While standard methods exist, they work best at coarse-grained national or state scales with abundant data, and struggle to accommodate the partial observability and sparse data common at finer scales (e.g., individual schools or towns). For example, case counts may be sparse when only a small fraction of infections are caught by a testing program. Or, whether an infected individu…
This article introduces epidemia, an R package for Bayesian, regression-oriented modeling of infectious diseases. The implemented models define a likelihood for all observed data while also explicitly modeling transmission dynamics: an approach often termed as semi-mechanistic. Infections are propagated over time using renewal equations. This approach is inspired by self-exciting, continuous-time point processes such as the Hawkes process. A variety of inferential tasks can be performed using the package. Key epidemiological quantities, including reproduction numbers and latent infections, may…
Renewal processes are a popular approach used in modelling infectious disease outbreaks. In a renewal process, previous infections give rise to future infections. However, while this formulation seems sensible, its application to infectious disease can be difficult to justify from first principles. It has been shown from the seminal work of Bellman and Harris that the renewal equation arises as the expectation of an age-dependent branching process. In this paper we provide a detailed derivation of the original Bellman Harris process. We introduce generalisations, that allow for time-varying re…
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…
The reproduction number of deterministic models is an essential quantity to predict whether an epidemic will spread or die out. Thresholds for disease extinction contribute crucial knowledge on disease control, elimination, and mitigation of infectious diseases. Relationships between the basic reproduction numbers of two network-based ordinary differential equation vector-host models, and extinction thresholds of corresponding continuous-time Markov chain models are derived under some assumptions. Numerical simulation results for malaria and Rift Valley fever transmission on heterogeneous netw…
The basic reproduction number, $R_0$ is an important and widely used concept in the study of infectious diseases. We briefly review the recent trend of calculating the average of various $R_0$ estimates in systematic reviews aimed at estimating the basic reproduction number of SARS-CoV-2, and discuss the drawbacks and implications of using such averaging methods. Additionally, we argue that even a theoretically grounded approach such as next generation matrix could have practical impediments in its use. More generally, the practice of associating an infectious disease with a single value of $R…
A great issue of discussion of an infectious disease is its basic reproduction number R0, which provides an estimation of the contagiousness of the disease. When R0 > 1, the disease spread will potentially lead to an outbreak, such that of the ongoing COVID-19 pandemics. During the evolution of an outbreak, various non-pharmaceutical interventions are employed, the impact of which is frequently assessed by the reduction that they introduce to the effective reproduction number Rt; reduction below 1 is an indication of eventual dying out of the disease spread. Motivated by the fact that R0 essen…
Early estimates of the transmission potential of emerging and re-emerging infections are increasingly used to inform public health authorities on the level of risk posed by outbreaks. Existing methods to estimate the reproduction number generally assume exponential growth in case incidence in the first few disease generations, before susceptible depletion sets in. In reality, outbreaks can display sub-exponential (i.e., polynomial) growth in the first few disease generations, owing to clustering in contact patterns, spatial effects, inhomogeneous mixing, reactive behavior changes, or other mec…
A system of partial differential equations is derived as a model for the dynamics of a honey bee colony with a continuous age distribution, and the system is then extended to include the effects of a simplified infectious disease. In the disease-free case we analytically derive the equilibrium age distribution within the colony and propose a novel approach for determining the global asymptotic stability of a reduced model. Furthermore, we present a method for determining the basic reproduction number $R_0$ of the infection; the method can be applied to other age-structured disease models with …
During the first months, the Covid-19 pandemic has required most countries to implement complex sequences of non-pharmaceutical interventions, with the aim of controlling the transmission of the virus in the population. To be able to take rapid decisions, a detailed understanding of the current situation is necessary. Estimates of time-varying, instantaneous reproduction numbers represent a way to quantify the viral transmission in real time. They are often defined through a mathematical compartmental model of the epidemic, like a stochastic SEIR model, whose parameters must be estimated from …
Early estimates of the transmission properties of a newly emerged pathogen are critical to an effective public health response, and are often based on limited outbreak data. Here, we use simulations to investigate a potential source of bias in such estimates, arising from correlations between the viral load of cases in transmission chains. We show that this mechanism can affect estimates of fundamental transmission properties characterising the spread of a virus. Our computational model simulates a disease transmission mechanism in which the viral load of the infector at the time of transmissi…
In this paper, we propose a new model for the dynamics of COVID-19 infections. Our approach consists of seven phenotypes: the susceptible humans, exposed humans, infectious humans, the recovered humans, the quarantine population, the recovered-exposed and deceased population. We proved first through mathematical approach the positivity, boundness and existence of a solution to considered model. We also studied the existence of the disease free equilibrium and corresp onding stability. Hence, the actual reproduction number was calculated . We analyzed the dependence of basic reproductions R_0 o…
The basic reproduction number ($R_0$) is a threshold parameter for disease extinction or survival in isolated populations. However no human population is fully isolated from other human or animal populations. We use compartmental models to derive simple rules for the basic reproduction number for populations with local person-to-person transmission and exposure from some other source: either a reservoir exposure or imported cases. We introduce the idea of a reservoir-driven or importation-driven disease: diseases that would become extinct in the population of interest without reservoir exposur…
We show that the basic reproduction number of an SIS patch model with standard incidence is either strictly decreasing and strictly convex with respect to the diffusion coefficient of infected subpopulation if the patch reproduction numbers of at least two patches in isolation are distinct or constant otherwise. Biologically, it means that fast diffusion of the infected people reduces the risk of infection. This completely solves and generalizes a conjecture by Allen et al. ({\it SIAM J Appl Math}, 67: 1283-1309, 2007). Furthermore, a substantially improved lower bound on the multipatch reprod…
This work presents a model combining the simplest communicable and non-communicable disease models. The latter is, by far, the leading cause of sickness and death in the World, and introduces basal heterogeneity in populations where communicable diseases evolve. The model can be interpreted as a risk-structured model, another way of accounting for population heterogeneity. Our results show that considering the non-communicable disease (in the end, heterogeneous populations) allows the communicable disease to become endemic even if the basic reproduction number is less than $1$. This feature is…
This is a pedagogical paper on estimating the number of people that can be infected by one infectious person during an epidemic outbreak, known as the reproduction number. Knowing the number is crucial for developing policy responses. There are generally two types of such a number, i.e., basic and effective (or instantaneous). While basic reproduction number is the average expected number of cases directly generated by one case in a population where all individuals are susceptible, effective reproduction number is the number of cases generated in the current state of a population. In this pape…
In this work we present a general framework for the modeling of the transmission dynamics of macroparasites which do not reproduce within the host like Ascaris lumbricoides, Trichuris trichiura, Necator americanus y Ancylostoma duodenale. The basic models are derived from general probabilistic models for the parasite density-dependent mating probability. Here we considered the particular, and common case, of a negative binomial distribution for the number of parasites in hosts. We find the basic reproductive number and we show that the system exhibit a saddle-node bifurcation at some value of …
Vector-transmitted diseases such as dengue fever and chikungunya have been spreading rapidly in many parts of the world. The Zika virus has been known since 1947 and invaded South America in 2013. It can be transmitted not only by (mosquito) vectors but also directly through sexual contact. Zika has developed into a serious global health problem because, while most cases are asymptomatic or very light, babies born to Zika - infected mothers may develop microcephaly and other very serious birth defects. We formulate and analyze two epidemic models for vector-transmitted diseases, one appropriat…
We propose a compartmental mathematical model for the spread of the COVID-19 disease with special focus on the transmissibility of super-spreaders individuals. We compute the basic reproduction number threshold, we study the local stability of the disease free equilibrium in terms of the basic reproduction number, and we investigate the sensitivity of the model with respect to the variation of each one of its parameters. Numerical simulations show the suitability of the proposed COVID-19 model for the outbreak that occurred in Wuhan, China.
The ongoing pandemic of Coronavirus disease (COVID-19) emerged in Wuhan, China in the end of 2019. It has already affected more than 300,000 people, with the number of deaths nearing 13000 across the world. As it has been posing a huge threat to global public health, it is of utmost importance to identify the rate at which the disease is spreading. In this study, we propose a time series model to analyze the trend pattern of the incidence of COVID-19 outbreak. We also incorporate information on total or partial lockdown, wherever available, into the model. The model is concise in structure, an…
Dengue is a vector-borne disease transmitted from an infected human to an Aedes mosquito, during a blood-meal. Dengue is still a major public health problem. A model for the disease transmission is presented, composed by human and mosquitoes compartments. The aim is to simulate the effects of seasonality, on the vectorial capacity and, consequently, on the disease development. Using entomological information about the mosquito behavior under different temperatures and rainfall, simulations are carried out and the repercussions analyzed. The basic reproduction number of the model is given, as w…
To control the current outbreak of the Coronavirus Disease 2019, constant monitoring of the epidemic is required since, as of today, no vaccines or antiviral drugs against it are known. We provide daily updated estimates of the reproduction number over time at https://stochastik-tu-ilmenau.github.io/COVID-19/. In this document, we describe the estimator we are using which was developed in (Fraser 2007), derive its asymptotic properties, and we give details on its implementation. Furthermore, we validate the estimator on simulated data, demonstrate that estimates on real data lead to plausible …
We develop a multi-patch and multi-group model that captures the dynamics of an infectious disease when the host is structured into an arbitrary number of groups and interacts into an arbitrary number of patches where the infection takes place. In this framework, we model host mobility that depends on its epidemiological status, by a Lagrangian approach. This framework is applied to a general SEIRS model and the basic reproduction number $\mathcal{R_0}$ is derived. The effects of heterogeneity in groups, patches and mobility patterns on $\mathcal{R_0}$ and disease prevalence are explored. Our …
Objectives.--To estimate the basic reproduction number of the Wuhan novel coronavirus (2019-nCoV). Methods.--Based on the susceptible-exposed-infected-removed (SEIR) compartment model and the assumption that the infectious cases with symptoms occurred before January 25, 2020 are resulted from free propagation without intervention, we estimate the basic reproduction number of 2019-nCoV according to the reported confirmed cases and suspected cases, as well as the theoretical estimated number of infected cases by other research teams, together with some epidemiological determinants learned from t…
We examine how spatial heterogeneity combines with mobility network structure to influence vector-borne disease dynamics. Specifically, we consider a Ross-Macdonald-type disease model on $n$ spatial locations that are coupled by host movement on a strongly connected, weighted, directed graph. We derive a closed form approximation to the domain reproduction number using a Laurent series expansion, and use this approximation to compute sensitivities of the basic reproduction number to model parameters. To illustrate how these results can be used to help inform mitigation strategies, as a case st…
We propose a population model for HIV-TB co-infection dynamics by considering treatments for HIV infection, active tuberculosis and co-infection. The HIV only and TB only models are analyzed separately, as well as full model. The basic reproduction numbers for TB ($\mathcal{R}_0^T$) and HIV ($\mathcal{R}_0^H$) and overall reproduction number for the system $\mathcal{R}_0= \max\{\mathcal{R}_0^T, \mathcal{R}_0^H\}$ are computed. The equilibria and their stability are studied. The main model undergoes supercritical transcritical bifurcation at $\mathcal{R}_0^T=1$ and $\mathcal{R}_0^H=1$ whereas t…
Addressed in this work is the performance of five popular algorithms, which aim at assessing the dissemination dynamics of the COVID-19 disease on the basis of the time series of new confirmed cases. The tests are based on simulated data, generated by means of a deterministic compartmental epidemiological model \cite{Matsinos2020a}, adapted herein to also include the possibility of the loss of immunity by the group of the recovered (or vaccinated) subjects. Assuming a simple temporal dependence of the effective reproduction number (the exact details are of no relevance as far as the conclusion…
The coronavirus pandemic has rapidly evolved into an unprecedented crisis. The susceptible-infectious-removed (SIR) model and its variants have been used for modeling the pandemic. However, time-independent parameters in the classical models may not capture the dynamic transmission and removal processes, governed by virus containment strategies taken at various phases of the epidemic. Moreover, very few models account for possible inaccuracies of the reported cases. We propose a Poisson model with time-dependent transmission and removal rates to account for possible random errors in reporting …
The basic reproduction number R0 -- the number of individuals directly infected by an infectious person in an otherwise susceptible population -- is arguably the most widely used estimator of how severe an epidemic outbreak can be. This severity can be more directly measured as the fraction people infected once the outbreak is over, Ω. In traditional mathematical epidemiology and common formulations of static network epidemiology, there is a deterministic relationship between R0 and Ω. However, if one considers disease spreading on a temporal contact network -- where one knows when contacts ha…
We will study a mathematical model of the human immunodeficiency virus (HIV) infection in the presence of combination therapy that includes within-host infectious dynamics. The deterministic model requires us to analyze asymptotic stability of two distinct steady states, disease-free and endemic equilibria. Previous results have focused on investigating the global asymptotic stability of the trivial steady state using an implicit finite-difference method which generates a system of difference equations. We, instead, provide analytic solutions and long term attractive behavior for the endemic s…
During epidemics, estimation of the effective reproduction number (ERN) associated with infectious disease is a challenging topic for policy development and medical resource management. There is still an open question about the dominant factors to characterize in corona virus disease 2019 (COVID-19), although recent studies based on nonlinear regression with machine learning suggested mobility. The emergence of new viral variants is common in widespread pandemics. However, understanding the potential ERN of new variants is required for policy revision, including lockdown constraints. In this s…
The effective reproduction number, R(t), is a central point in the study of infectious diseases. It establishes in an explicit way the extent of an epidemic spread process in a population. The current estimation methods for the time evolution of R(t), using incidence data, rely on the generation interval distribution, g(τ), which is usually obtained from empirical data or already known distributions from the literature. However, there are systems, especially highly heterogeneous ones, in which there is a lack of data and an adequate methodology to obtain g(τ). In this work, we use mathematical…
This paper is an exploratory study of two epidemiological questions on a worldwide basis. How fast is the disease spreading? Are the restrictions (especially mobility restrictions) for people bring the expected effect? To answer the first question, we propose a tool for estimating the reproduction number of epidemic (the number of secondary infections $R_t$) based on the SEIR model and compare it with an non-model $R_t$ estimation. To measure the $R_t$ of COVID-19 for different countries, real-time data on coronavirus daily cases of infections, recoveries, deaths are retrieved from the Center …
Recent evidences show that individuals who recovered from COVID-19 can be reinfected. However, this phenomenon has rarely been studied using mathematical models. In this paper, we propose a SEIRE epidemic model to describe the spread of the epidemic with reinfection. We obtain the important thresholds $R_0$ (the basic reproduction number) and Rc (a threshold less than one). Our investigations show that when $R_0 > 1$, the system has an endemic equilibrium, which is globally asymptotically stable. When $R_c < R_0 < 1$, the epidemic system exhibits bistable dynamics. That is, the system has back…
A mathematical model was developed describing the dynamic of the COVID-19 virus over a population considering that the infected can either be symptomatic or not. The model was calibrated using data on the confirmed cases and death from several countries like France, Philippines, Italy, Spain, United Kingdom, China, and the USA. First, we derived the basic reproduction number, $\mathcal{R}_0$, and estimated the effective reproduction $\mathcal{R}_{\rm eff}$ for each country. Second, we were interested in the merits of interventions, either by distancing or by treatment. Results revealed that to…
Background: In 2015, the Zika arbovirus (ZIKV) began circulating in the Americas, rapidly expanding its global geographic range in explosive outbreaks. Unusual among mosquito-borne diseases, ZIKV has been shown to also be sexually transmitted, although sustained autochthonous transmission due to sexual transmission alone has not been observed, indicating the reproduction number (R0) for sexual transmission alone is less than 1. Critical to the assessment of outbreak risk, estimation of the potential attack rates, and assessment of control measures, are estimates of the basic reproduction numbe…
In this paper, we study the psychological effect in a SIS epidemic model. The basic reproduction number is obtained. However, the disease free equilibrium is always asymptotically stable, which doesn't depends on the basic reproduction number. The system has a saddle-node bifurcation appear and displays bistable behavior, which is a new phenomenon in epidemic dynamics and different from the backward bifurcation behavior.
We study the global dynamics of a reaction-diffusion SEIR infection model with distributed delay and nonlinear incidence rate. The well-posedness of the proposed model is proved. By means of Lyapunov functionals, we show that the disease free equilibrium state is globally asymptotically stable when the basic reproduction number is less or equal than one, and that the disease endemic equilibrium is globally asymptotically stable when the basic reproduction number is greater than one. Numerical simulations are provided to illustrate the obtained theoretical results.
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 distribu…
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…
Many studies have proposed a relationship between COVID-19 transmissibility and ambient pollution levels. However, a major limitation in establishing such associations is to adequately account for complex disease dynamics, influenced by e.g. significant differences in control measures and testing policies. Another difficulty is appropriately controlling the effects of other potentially important factors, due to both their mutual correlations and a limited dataset. To overcome these difficulties, we will here use the basic reproduction number ($R_0$) that we estimate for USA states using non-li…
