We consider a single outbreak susceptible-infected-recovered (SIR) model and corresponding estimation procedures for the effective reproductive number $\mathcal{R}(t)$. We discuss the estimation of the underlying SIR parameters with a generalized least squares (GLS) estimation technique. We do this in the context of appropriate statistical models for the measurement process. We use asymptotic statistical theories to derive the mean and variance of the limiting (Gaussian) sampling distribution and to perform post statistical analysis of the inverse problems. We illustrate the ideas and pitfalls (e.g., large condition numbers on the corresponding Fisher information matrix) with both synthetic and influenza incidence data sets.
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…
In Costa Rica, the first known cases of Zika were reported in 2016. We looked at the 2016-2017 Zika outbreak and explored the transmission dynamics using weekly reported data. A nonlinear differential equation single-outbreak model with sexual transmission, as well as host availability for vector-feeding was used to estimate key parameters, fit the data and compute the {\it basic reproductive number}, $\mathcal{R}_0$, distribution. Furthermore, a sensitivity and elasticity analysis was computed based on the $\mathcal{R}_0$ parameters.
Vector-borne diseases with reservoir cycles are complex to understand because new infections come from contacts of the vector with humans and different reservoirs. In this scenario, the basic reproductive number $\mathcal{R}^h_0$ of the system where the reservoirs are not included could turn out to be less than one, yet, an endemic equilibrium be observed. Indeed, when the reservoirs are taken back into account, the basic reproductive number $\mathcal{R}_0^r$, of only vectors and reservoirs, explains the endemic state. Furthermore, reservoirs cycles with a small basic reproductive number could…
Summary 1. Infectious disease outbreaks in plants threaten ecosystems, agricultural crops and food trade. Currently, several fungal diseases are affecting forests worldwide, posing a major risk to tree species, habitats and consequently ecosystem decay. Prediction and control of disease spread are difficult, mainly due to the complexity of the interaction between individual components involved. 2. In this work, we introduce a lattice-based epidemic model coupled with a stochastic process that mimics, in a very simplified way, the interaction between the hosts and pathogen. We studied the disea…
Since the beginning of March 2020, the cumulative numbers of cases of infection with the novel coronavirus SARS-CoV-2 in Germany have been reported on a daily basis. The reports originate from national laws, according to which positive test findings must be submitted to the Federal Health Authorities, the Robert Koch Institute, via the local health authorities. Since an enormous number of unreported cases can be expected, the question of how widespread the disease has been in the population cannot be answered based on these administrative reports. Using mathematical modeling, however, estimate…
With the unfolding of the COVID-19 pandemic, mathematical modeling of epidemics has been perceived and used as a central element in understanding, predicting, and governing the pandemic event. However, soon it became clear that long term predictions were extremely challenging to address. Moreover, it is still unclear which metric shall be used for a global description of the evolution of the outbreaks. Yet a robust modeling of pandemic dynamics and a consistent choice of the transmission metric is crucial for an in-depth understanding of the macroscopic phenomenology and better-informed mitiga…
We present a compartmental mathematical model with demography for the spread of the COVID-19 disease, considering also asymptomatic infectious individuals. We compute the basic reproductive ratio of the model and study the local and global stability for it. We solve the model numerically based on the case of Italy. We propose a vaccination model and we derive threshold conditions for preventing infection spread in the case of imperfect vaccines.
Background: A deterministic model is developed for the spatial spread of an epidemic disease in a geographical setting. The disease is borne by vectors to susceptible hosts through criss-cross dynamics. The model is focused on an epidemic outbreak that initiates from a small number of cases in a small sub-region of the geographical setting. Methods: Partial differential equations are formulated to describe the interaction of the model compartments. Results: The partial differential equations of the model are analyzed and proven to be well-posed. The epidemic outcomes of the model are correlate…
Epidemic spread in single-host systems strongly depends on the population's contact network. However, little is known regarding the spread of epidemics across networks representing populations of multiple hosts. We explored cross-species transmission in a multilayer network where layers represent populations of two distinct hosts, and disease can spread across intralayer (within-host) and interlayer (between-host) edges. We developed an analytic framework for the SIR epidemic model to examine the effect of (i) source of infection and (ii) between-host asymmetry in infection probabilities, on d…
As of October 12, 2018, 211 cases of Ebola virus disease (EVD) were reported in North Kivu Province, Democratic Republic of Congo. Since the beginning of October the outbreak has largely shifted into regions in which active armed conflict is occurring, and in which EVD cases and their contacts are difficult for health workers to reach. We modeled EVD transmission using a branching process with gradually quenching transmission estimated from past EVD outbreaks, with outbreak trajectories conditioned on agreement with the course of the current outbreak, and with multiple levels of vaccination co…
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…
We consider the spread of a Susceptible-Infected-Recovered (SIR) disease through finite populations and derive an expression for the final size distribution. Our derivation allows arbitrary distributions of the number of transmissions caused by an infected individual. We show how this calculation can be used to infer parameters of the infectious disease through observations in multiple small populations. The inference suffers from some identifiability difficulties, and it requires many observations to distinguish between parameter combinations that correspond to the same reproductive number.
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…
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…
The commonly observed negative correlation between the number of species in an ecological community and disease risk, typically referred to as "the dilution effect", has received a substantial amount of attention over the past decade. Attempts to test this relationship experimentally have revealed that, in addition to the mean disease risk decreasing with species number, so too does the variance of disease risk. This is referred to as the "variance reduction effect", and has received relatively little attention in the disease-diversity literature. Here, we set out to clarify and quantify some …
The Ebola virus in West Africa has infected almost 30,000 and killed over 11,000 people. Recent models of Ebola Virus Disease (EVD) have often made assumptions about how the disease spreads, such as uniform transmissibility and homogeneous mixing within a population. In this paper, we test whether these assumptions are necessarily correct, and offer simple solutions that may improve disease model accuracy. First, we use data and models of West African migration to show that EVD does not homogeneously mix, but spreads in a predictable manner. Next, we estimate the initial growth rate of EVD wit…
COVID-19 pandemic has become a major threat to the country. Till date, well tested medication or antidote is not available to cure this disease. According to WHO reports, COVID-19 is a severe acute respiratory syndrome which is transmitted through respiratory droplets and contact routes. Analysis of this disease requires major attention by the Government to take necessary steps in reducing the effect of this global pandemic. In this study, outbreak of this disease has been analysed for India till 30th March 2020 and predictions have been made for the number of cases for the next 2 weeks. SEIR …
Introduction: Zika epidemic in America was declared a public health emergency of international concern after the rapid spread in the region. Stratification of the potential transmission of the disease is needed to address the efforts surveillance and disease control. The goal of this research is to compare the basic reproductive number of Zika in different municipalities, from an SIR model with implicit vector dynamics, based on the daily case reporting data of Antioquia, Colombia, the second most affected country after Brazil. Methods: An simple SIR model with implicit vector dynamics was der…
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…
We use a multivariate formulation of sequential Monte Carlo filter that utilizes mechanistic models for Ebola virus propagation and available incidence data to simultaneously estimate the disease progression states and the model parameters. This method has the advantage of performing the inference online as the new data becomes available and estimates the evolution of basic reproductive ratio $R_0(t)$ of the Ebola outbreak through time. Our analysis identifies a peak in the basic reproductive ratio close to the time when Ebola cases were reported in Europe and the USA.
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…
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…
From footpaths to flight routes, human mobility networks facilitate the spread of communicable diseases. Control and elimination efforts depend on characterizing these networks in terms of connections and flux rates of individuals between contact nodes. In some cases, transport can be parameterized with gravity-type models or approximated by a diffusive random walk. As a alternative, we have isolated intranational commercial air traffic as a case study for the utility of non-diffusive, heavy-tailed transport models. We implemented new stochastic simulations of a prototypical influenza-like inf…
We develop open source infection risk calculators for patients and healthcare professionals as apps for hospital acquired infections (during child-delivery) and sexually transmitted infections (like HIV). Advanced versions of ehealth in non-communicable diseases do not apply to epidemiology much. There is, however, no infection risk calculator in the Polish Internet so far, despite the existence of data that may be applied to create such a tool. The algorithms involve data from Information Systems (like HIS in hospitals) and surveys by applying mathematical modelling, Bayesian inference, logis…
Heidelberg University（海德堡大学）；Chinese Academy of Medical Sciences and Peking Union Medical College（中国医学科学院与北京协和医学院）；University Hospital Bonn（波恩大学医院）；University Hospital Heidelberg（海德堡大学医院）；Harvard T.H. Chan School of Public Health（哈佛大学陈曾熙公共卫生学院）；Africa Health Research Institute（非洲健康研究所）
Mathematical models in epidemiology are an indispensable tool to determine the dynamics and important characteristics of infectious diseases. Apart from their scientific merit, these models are often used to inform political decisions and intervention measures during an ongoing outbreak. However, reliably inferring the dynamics of ongoing outbreaks by connecting complex models to real data is still hard and requires either laborious manual parameter fitting or expensive optimization methods which have to be repeated from scratch for every application of a given model. In this work, we address …
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 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…
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…
Without vaccines and treatments, societies must rely on non-pharmaceutical intervention strategies to control the spread of emerging diseases such as COVID-19. Though complete lockdown is epidemiologically effective, because it eliminates infectious contacts, it comes with significant costs. Several recent studies have suggested that a plausible compromise strategy for minimizing epidemic risk is periodic closure, in which populations oscillate between wide-spread social restrictions and relaxation. However, no underlying theory has been proposed to predict and explain optimal closure periods …
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…
There has been a large number of reported cases of the occurrence of Zika in different countries in 2016 and it is necessary to develop an early warning system to initiate preventive campaigns against the disease. A potential early warning system based on the rise in ocean temperature of the Pacific Niño Index is proposed. The efficacy is verified using data for the outbreak in Colombia as obtained from Google Trends.
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）
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…
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 …
