We discuss the generation of various reproduction ratios or numbers that are very useful to monitor an ongoing epidemic like Covid-19 and examine the effects of intervention measures. A detailed SEIR algorithm is described for their computation, with applications given to the current Covid-19 outbreaks in a number of countries (Argentina, Brazil, France, Italy, Mexico, Spain, UK and USA). The corresponding matlab script, complete and ready to use, is provided for free downloading.
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…
The 2014 Ebola virus (EBOV) outbreak in West Africa is the largest outbreak of the genus Ebolavirus to date. To better understand the spread of infection in the affected countries, it is crucial to know the number of secondary cases generated by an infected index case in the absence and presence of control measures, i.e., the basic and effective reproduction number. In this study, I describe the EBOV epidemic using an SEIR (susceptible-exposed-infectious-recovered) model and fit the model to the most recent reported data of infected cases and deaths in Guinea, Sierra Leone and Liberia. The max…
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…
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 work we analyze the post monsoon Dengue outbreaks by analyzing the transient and long term dynamics of Dengue incidences and its environmental correlates in Ahmedabad city in western India from 2005-2012. We calculate the reproduction number $R_p$ using the growth rate of post monsoon Dengue outbreaks and biological parameters like host and vector incubation periods and vector mortality rate, and its uncertainties are estimated through Monte-Carlo simulations by sampling parameters from their respective probability distributions. Reduction in Female Aedes mosquito density required for …
In the absence of other tools, monitoring the effects of protective measures, including social distancing and forecasting the outcome of outbreaks is of immense interest. Real-time data is noisy and very often hampered by systematic errors in reporting. Detailed epidemic models may contain a large number of empirical parameters, which cannot be determined with sufficient accuracy. In this paper, we show that the cumulative number of deaths can be regarded as a master variable, and the parameters of the epidemic such as the basic reproduction number, the size of the susceptible population, and …
The recent outbreak of COVID-19 in Mainland China is characterized by a distinctive algebraic, sub-exponential increase of confirmed cases during the early phase of the epidemic, contrasting an initial exponential growth expected for an unconstrained outbreak with sufficiently large reproduction rate. Although case counts vary significantly between affected provinces in Mainland China, the scaling law $t^μ$ is surprisingly universal, with a range of exponents $μ=2.1\pm0.3$. The universality of this behavior indicates that despite social, regional, demographical, geographical, and socio-economi…
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…
Based on the reported data until 18 March 2015 and numerical fitting via a simple formula of cumulative case number, we provide real-time estimation on basic reproduction number, inflection point, peak time and final outbreak size of ongoing Ebola outbreak in West Africa. From our simulation, we conclude that the first wave has passed its inflection point and predict that a second epidemic wave may appear in the near future.
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…
We consider the SIR epidemiological model applied to the evolution of COVID-19 with two approaches. In the first place we fit a global SIR model, with time delay, and constant parameters throughout the outbreak, including the contagion rate. The contention measures are reflected on an effective reduced susceptible population $N_{eff}$. In the second approach we consider a time-dependent contagion rate that reflects the contention measures either through a step by step fitting process or by following an exponential decay. In this last model the population is considered the one of the country $N…
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…
As the interactions between people increases, the impending menace of COVID-19 outbreaks materialize, and there is an inclination to apply lockdowns. In this context, it is essential to have easy-to-use indicators for people to use as a reference. The basic reproduction number of confirmed positives, $R_t$, fulfill such a role. This document proposes a data-driven approach to nowcast $R_t$ based on previous observations' statistical behavior. As more information arrives, the method naturally becomes more precise about the final count of confirmed positives. Our method's strength is that it is …
We introduce a theoretical framework that highlights the impact of physical distancing variables such as human mobility and physical proximity on the evolution of epidemics and, crucially, on the reproduction number. In particular, in response to the coronavirus disease (CoViD-19) pandemic, countries have introduced various levels of 'lockdown' to reduce the number of new infections. Specifically we use a collisional approach to an infection-age structured model described by a renewal equation for the time homogeneous evolution of epidemics. As a result, we show how various contributions of th…
In the night of February 20, 2020, the first case of novel coronavirus disease (COVID-19) was confirmed in the Lombardy Region, Italy. In the week that followed, Lombardy experienced a very rapid increase in the number of cases. We analyzed the first 5,830 laboratory-confirmed cases to provide the first epidemiological characterization of a COVID-19 outbreak in a Western Country. Epidemiological data were collected through standardized interviews of confirmed cases and their close contacts. We collected demographic backgrounds, dates of symptom onset, clinical features, respiratory tract speci…
In the absence of neither an effective treatment or vaccine and with an incomplete understanding of the epidemiological cycle, Govt. has implemented a nationwide lockdown to reduce COVID-19 transmission in India. To study the effect of social distancing measure, we considered a new mathematical model on COVID-19 that incorporates lockdown effect. By validating our model to the data on notified cases from five different states and overall India, we estimated several epidemiologically important parameters as well as the basic reproduction number ($R_{0}$). Combining the mechanistic mathematical …
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…
We have further extended our compartmental model describing the spread of the infection in Italy. The model is based on the assumption that the time evolution of all of the observable quantities (number of people still positive to the infection, hospitalized and fatalities cases, healed people, and total number of people that has contracted the infection) depend on average parameters, namely people diffusion coefficient, infection cross-section, and population density. The model provides precious information on the tight relationship between the variation of the reported infection cases and a …
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…
Analytical expressions for the basic reproduction number, R0, have been obtained in the past for a wide variety of mathematical models for infectious disease spread, along with expressions for the expected final size of an outbreak. However, what has so far not been studied is the average number of infections that descend down the chains of infection begun by each of the individuals infected in an outbreak (we refer to this quantity as the "average number of descendant infections" per infectious individual, or ANDI). ANDI includes not only the number of people that an individual directly conta…
We study the datafor the cumulative as well as daily number of cases in the Covid-19 outbreak in China. The cumulative data can be fit to an empirical form obtained from a Susceptible-Infected-Removed (SIR) model studied on an Euclidean network previously. Plotting the number of cases against the distance from the epicenter for both China and Italy, we find an approximate power law variation with an exponent $\sim 1.85$ showing strongly that the spatial dependence plays a key role, a factor included in the model. We report here that the SIR model on the Eucledean network can reproduce with a h…
We use daily data from Lombardy, the Italian region most affected by the COVID-19 outbreak, to calibrate a SIR model individually on each municipality. These are all covered by the same health system and, in the post-lockdown phase we focus on, all subject to the same social distancing regulations. We find that municipalities with a higher number of cases at the beginning of the period analyzed have a lower rate of diffusion, which cannot be imputed to herd immunity. In particular, there is a robust and strongly significant negative correlation between the estimated basic reproduction number (…
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.
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…
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 …
The spread of pandemics such as COVID-19 is strongly linked to human activities. The objective of this paper is to specify and examine early indicators of disease spread risk in cities during the initial stages of outbreak based on patterns of human activities obtained from digital trace data. In this study, the Venables distance (D_v), and the activity density (D_a) are used to quantify and evaluate human activities for 193 US counties, whose cumulative number of confirmed cases was greater than 100 as of March 31, 2020. Venables distance provides a measure of the agglomeration of the level o…
Contact tracing is an important control strategy for containing Ebola epidemics. From a theoretical perspective, explicitly incorporating contact tracing with disease dynamics presents challenges, and population level effects of contact tracing are difficult to determine. In this work, we formulate and analyze a mechanistic SEIR type outbreak model which considers the key features of contact tracing, and we characterize the impact of contact tracing on the effective reproduction number, $\mathcal R_e$, of Ebola. In particular, we determine how relevant epidemiological properties such as incuba…
The paper deals with a modeling of the ongoing epidemic caused by Coronavirus disease 2019 (COVID-19) on the closed territory of the Republic of Croatia. Using the official public information on the number of confirmed infected, recovered and deceased individuals, the modified SEIR compartmental model is developed to describe the underlying dynamics of the epidemic. Fitted modified SEIR model provides the prediction of the disease progression in the near future, considering strict control interventions by means of social distancing and quarantine for infected and at-risk individuals introduced…
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…
We propose and analyse a mathematical model for cholera considering vaccination. We show that the model is epidemiologically and mathematically well posed and prove the existence and uniqueness of disease-free and endemic equilibrium points. The basic reproduction number is determined and the local asymptotic stability of equilibria is studied. The biggest cholera outbreak of world's history began on 27th April 2017, in Yemen. Between 27th April 2017 and 15th April 2018 there were 2275 deaths due to this epidemic. A vaccination campaign began on 6th May 2018 and ended on 15th May 2018. We show…
An outbreak of respiratory disease caused by a novel coronavirus is ongoing from December 2019. As of July 22, 2020, it has caused an epidemic outbreak with more than 15 million confirmed infections and above 6 hundred thousand reported deaths worldwide. During this period of an epidemic when human-to-human transmission is established and reported cases of coronavirus disease 2019 (COVID-19) are rising worldwide, investigation of control strategies and forecasting are necessary for health care planning. In this study, we propose and analyze a compartmental epidemic model of COVID-19 to predict…
Modeling Epidemics on Multiplex Networks: Epidemic Threshold and Basic Reproduction Number
Instituto Balseiro, Universidad Nacional de Cuyo（昆都萨国立大学巴尔塞罗研究所）；Centro Atómico Bariloche, Comisión Nacional de Energía Atómica（阿根廷国家原子能委员会巴里洛切原子中心）；Consejo Nacional de Investigaciones Científicas y Técnicas (CONICET)（阿根廷国家科学研究委员会）；Universidad Nacional del Centro de la Provincia de Buenos Aires, Facultad de Cs. Exactas, Instituto Multidisciplinario Sobre Ecosistemas Y Desarrollo Sustentable（布宜诺斯艾利斯省中部国立大学理学院跨学科生态系统与可持续发展研究所）；Universidad Nacional del Centro de la Provincia de Buenos Aires, Facultad de Cs. Exactas, Instituto PLADEMA（布宜诺斯艾利斯省中部国立大学理学院PLADEMA研究所）
In order to analyze the effectiveness of three successive nationwide lockdown enforced in India, we present a data-driven analysis of four key parameters, reducing the transmission rate, restraining the growth rate, flattening the epidemic curve and improving the health care system. These were quantified by the consideration of four different metrics, namely, reproduction rate, growth rate, doubling time and death to recovery ratio. The incidence data of the COVID-19 (during the period of 2nd March 2020 to 31st May 2020) outbreak in India was analyzed for the best fit to the epidemic curve, ma…
As South and Central American countries prepare for increased birth defects from Zika virus outbreaks and plan for mitigation strategies to minimize ongoing and future outbreaks, understanding important characteristics of Zika outbreaks and how they vary across regions is a challenging and important problem. We developed a mathematical model for the 2015 Zika virus outbreak dynamics in Colombia, El Salvador, and Suriname. We fit the model to publicly available data provided by the Pan American Health Organization, using Approximate Bayesian Computation to estimate parameter distributions and p…
The outbreak of corona virus disease 2019 (COVID-19), caused by the virus severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2) has already created emergency situations in almost every country of the world. The disease spreads all over the world within a very short period of time after its first identification in Wuhan, China in December, 2019. In India, the outbreaks starts on $2^{nd}$ March, 2020 and after that the cases are increasing exponentially. Very high population density, the unavailability of specific medicines or vaccines, insufficient evidences regarding the transmission me…
A dynamical epidemic model optimized using genetic algorithm and cross validation method to overcome the overfitting problem is proposed. The cross validation procedure is applied so that available data are split into a training subset used to fit the algorithm's parameters, and a smaller subset used for validation. This process is tested on the countries of Italy, Spain, Germany and South Korea before being applied to Algeria. Interestingly, our study reveals an inverse relationship between the size of the training sample and the number of generations required in the genetic algorithm. Moreov…
A general stochastic model for susceptible -> infective -> recovered (SIR) epidemics in non homogeneous populations is considered. The heterogeneity is a very important aspect here since it allows more realistic but also more complex models. The basic reproduction number $R_0$, an indication of the probability of an outbreak for homogeneous populations does not indicate the probability of an outbreak for non homogeneous models anymore, because it changes with the initially infected case. Therefore, we use "individual $R_0$" that is the expected number of secondary cases for a given initially i…
Viral outbreaks, such as the current COVID-19 pandemic, are commonly described by compartmental models by means of ordinary differential equation (ODE) systems. The parameter values of these ODE models are typically unknown and need to be estimated based on accessible data. In order to describe realistic pandemic scenarios with strongly varying situations, these model parameters need to be assumed as time-dependent. While parameter estimation for the typical case of time-constant parameters does not pose larger issues, the determination of time-dependent parameters, e.g.~the transition rates o…
Background: Northern Italy was one of the most impacted areas by COVID. It is now widely assumed that the virus was silently spreading for at least 2 weeks before the first patient was identified. During this silent phase, and in the following weeks when the hospital system was overburdened, data collection was not performed in an accurate enough way to estimate an epidemic curve. With the aim of assessing both the dynamics of the introduction of the virus and the effectiveness of containment measures introduced, we try to reconstruct the epidemic curve using all cause mortality data. Methods:…
We study the reported data from the COVID-19 pandemic outbreak in January - May 2020 in 119 countries. We observe that the time series of active cases in individual countries (the difference of the total number of confirmed infections and the sum of the total number of reported deaths and recovered cases) display a strong agreement with polynomial growth and at a later epidemic stage also with a combined polynomial growth with exponential decay. Our results are also formulated in terms of compartment type mathematical models of epidemics. Within these models the universal scaling characterizin…
We propose a mathematical model for the transmission dynamics of some strains of the bacterium Vibrio cholerae, responsible for the cholera disease in humans. We prove that, when the basic reproduction number is equal to one, a transcritical bifurcation occurs for which the endemic equilibrium emanates from the disease-free point. A control function is introduced into the model, representing the distribution of chlorine water tablets for water purification. An optimal control problem is then proposed and analyzed, where the goal is to determine the fraction of susceptible individuals who shoul…
Bayesian Inference of Reproduction Number from Epidemiological and Genetic Data Using Particle MCMC
Nuffield Department of Medicine, University of Oxford（牛津大学纳菲尔德医学系）；School of Mathematics, Statistics and Physics, Newcastle University（纽卡斯尔大学数学、统计与物理学院）；Department of Statistics, University of Warwick（华威大学统计学系）；School of Life Sciences, University of Warwick（华威大学生命科学学院）；The Zeeman Institute for Systems Biology and Infectious Disease Epidemiology Research, University of Warwick（华威大学泽曼系统与传染病流行病学研究学院）
In epidemiology, the effective reproduction number $R_e$ is used to characterize the growth rate of an epidemic outbreak. In this paper, we investigate properties of $R_e$ for a modified SEIR model of COVID-19 in the city of Houston, TX USA, in which the population is divided into low-risk and high-risk subpopulations. The response of $R_e$ to two types of control measures (testing and distancing) applied to the two different subpopulations is characterized. A nonlinear cost model is used for control measures, to include the effects of diminishing returns. We propose three types of heuristic s…
There are several reports in India that indicate hospitals and quarantined centers are COVID-19 hotspots. In the absence of efficient contact tracing tools, Govt. and the policymakers may not be paying attention to the risk of hospital-based transmission. To explore more on this important route and its possible impact on lockdown effect, we developed a mechanistic model with hospital-based transmission. Using daily notified COVID-19 cases from six states (Maharashtra, Delhi, Madhya Pradesh, Rajasthan, Gujarat, and Uttar Pradesh) and overall India, we estimated several important parameters of t…
Background: The ongoing COVID-19 epidemic dilated rapidly throughout India. To end the global COVID-19 pandemic major behavioral, social distancing, contact tracing, and state interventions has been undertaken to reduce the outbreak and avert the persistence of the coronavirus in humans in India and worldwide. In absence of any vaccine or therapeutics, forecasting is of utmost priority for health care planning and control the transmission of COVID-19. Methods: We have proposed a mathematical model that explain the transmission dynamics of COVID-19 in India. Based on the estimated data our mode…
Infectious disease transmission dynamics are particularly sensitive to social contact patterns, and the precautions people take to limit disease transmission. It depends on the age distribution of the community. Thus, knowing the age$-$specific prevalence and incidence of infectious diseases is critical for predicting future disease burden and the efficacy of interventions like immunization. This study uses an SEIR age$-$structured multi-group epidemic model to understand how social contact affects disease control. We created location$-$specific social contact matrices in the community to see …
We propose an SEIARD mathematical model to investigate the current outbreak of coronavirus disease (COVID-19) in Mexico. We conduct a detailed analysis of this model and demonstrate its application using publicly reported data. We calculate the basic reproduction number ($R_0$) via the next-generation matrix method, and we estimate the per day infection, death and recovery rates. We calibrate the parameters of the SEIARD model to the reported data by minimizing the sum of squared errors and attempt to forecast the evolution of the outbreak until June 2020. Our results estimate that the peak of…
For nearly a century, the initial reproduction number (R0) has been used as a one number summary to compare outbreaks of infectious disease, yet there is no `standard' estimator for R0. Difficulties in estimating R0 arise both from how a disease transmits through a population as well as from differences in statistical estimation method. We describe eight methods used to estimate R0 and provide a thorough simulation study of how these estimates change in the presence of different disease parameters. As motivation, we analyze the 2009 outbreak of the H1N1 pandemic influenza in the USA and compar…
SARS-CoV-2 causing COVID-19 disease has moved rapidly around the globe, infecting millions and killing hundreds of thousands. The basic reproduction number, which has been widely used and misused to characterize the transmissibility of the virus, hides the fact that transmission is stochastic, is dominated by a small number of individuals, and is driven by super-spreading events (SSEs). The distinct transmission features, such as high stochasticity under low prevalence, and the central role played by SSEs on transmission dynamics, should not be overlooked. Many explosive SSEs have occurred in …
During Covid-19 outbreaks, school closures are employed as part of governments' non-pharmaceutical interventions around the world to reduce the number of contacts and keep the reproduction number below 1. Yet, prolonged school closures have profound negative impact on the future opportunities of pupils, particularly from disadvantaged backgrounds, as well as additional economic and social impacts by preventing their parents from returning to work. Data on Covid-19 in children are sparse and policy frameworks are evolving quickly. We compare a set of potential policies to accompany the reopenin…
In late 2019, a novel coronavirus, the SARS-CoV-2 outbreak was identified in Wuhan, China and later spread to every corner of the globe. Whilst the number of infection-induced deaths in Ghana, West Africa are minimal when compared with the rest of the world, the impact on the local health service is still significant. Compartmental models are a useful framework for investigating transmission of diseases in societies. To understand how the infection will spread and how to limit the outbreak. We have developed a modified SEIR compartmental model with nine compartments (CoVCom9) to describe the d…
