The COVID-19 disease has forced countries to make a considerable collaborative effort between scientists and governments to provide indicators to suitable follow-up the pandemic's consequences. Mathematical modeling plays a crucial role in quantifying indicators describing diverse aspects of the pandemic. Consequently, this work aims to develop a clear, efficient, and reproducible methodology for parameter optimization, whose implementation is illustrated using data from three representative regions from Chile and a suitable generalized SIR model together with a fitted positivity rate. Our results reproduce the general trend of the infected's curve, distinguishing the reported and real cases. Finally, our methodology is robust, and it allows us to forecast a second outbreak of COVID-19 and the infection fatality rate of COVID-19 qualitatively according to the reported dead cases.
We first propose a quantitative approach to detect high risk outbreaks of independent and coinfective SIR dynamics on three empirical networks: a school, a conference and a hospital contact network. This measurement is based on the k-means clustering method and identifies proper samples for calculating the mean outbreak size and the outbreak probability. Then we systematically study the impact of different temporal correlations on high risk outbreaks over the original and differently shuffled counterparts of each network. We observe that, on the one hand, in the coinfection process, randomizat…
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…
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.
The aim of this work is to investigate the optimal vaccine sharing between two SIR centers in the presence of migration fluxes of susceptibles and infected individuals during the mumps outbreak. Optimality of the vaccine allocation means the minimization of the total number of lost working days during the whole period of epidemic outbreak $[0,t_f]$, which can be described by the functional $Q=\int_0^{t_f}I(t){\rm d}t$ where $I(t)$ stands for the number of infectives at time $t$. We explain the behavior of the optimal allocation, which depends on the model parameters and the amount of available…
In this paper we aim to study an optimal insurance premium level for health-care in a deterministic and stochastic SIR models with migration fluxes and vaccination of population. The studied model considers two standard SIR centres connected via links and continuous migration fluxes. The premium is calculated using the basic equivalence principle. Even in this simple setup there are non-intuitive results that illustrate how the premium depends on migration rates, severeness of a disease and initial distribution of healthy and infected individuals through the centres. We investigate how the vac…
It has been known that epidemic outbreaks in the SIR model on networks are described by phase transitions. Despite the similarity with percolation transitions, whether an epidemic outbreak occurs or not cannot be predicted with probability one in the thermodynamic limit. We elucidate its mechanism by deriving a simple Langevin equation that captures an essential aspect of the phenomenon. We also calculate the probability of epidemic outbreaks near the transition point.
In the Susceptible-Infectious-Recovered (SIR) model of disease spreading, the time to extinction of the epidemics happens at an intermediate value of the per-contact transmission probability. Too contagious infections burn out fast in the population. Infections that are not contagious enough die out before they spread to a large fraction of people. We characterize how the maximal extinction time in SIR simulations on networks depend on the network structure. For example we find that the average distances in isolated components, weighted by the component size, is a good predictor of the maximal…
Power-law scalings are ubiquitous to physical phenomena undergoing a continuous phase transition. The classic Susceptible-Infectious-Recovered (SIR) model of epidemics is one such example where the scaling behavior near a critical point has been studied extensively. In this system the distribution of outbreak sizes scales as $P(n) \sim n^{-3/2}$ at the critical point as the system size $N$ becomes infinite. The finite-size scaling laws for the outbreak size and duration are also well understood and characterized. In this work, we report scaling laws for a model with SIR structure coupled with …
We are currently facing a highly critical case of a world-wide pandemic. The novel coronavirus (SARS-CoV-2, a.k.a. COVID-19) has proved to be extremely contagious and the original outbreak from Asia has now spread to all continents. This situation will fruitfully profit from the study in regards of the spread of the virus, assessing effective countermeasures to weight the impact of the adopted strategies. The standard Susceptible-Infectious-Recovered (SIR) model is a very successful and widely used mathematical model for predicting the spread of an epidemic. We adopt the SIR model on a random …
The outbreak of Covid-19 led the world to an unprecedent health and economical crisis. In an attempt to responde to this emergency researchers worldwide are intensively studying the Covid-19 pandemic dynamics. In this work, a SIRSi compartmental model is proposed, which is a modification of the known classical SIR model. The proposed SIRSi model considers differences in the immunization within a population, and the possibility of unreported or asymptomatic cases. The model is adjusted to three major cities of São Paulo State, in Brazil, namely, São Paulo, Santos and Campinas, providing estimat…
Containment measures implemented by some countries to suppress the spread of COVID-19 have resulted in a slowdown of the epidemic characterized by time series of daily infections plateauing over extended periods of time. We prove that such a dynamical pattern is compatible with critical Susceptible-Infected-Removed (SIR) dynamics. In traditional analyses of the critical SIR model, the critical dynamical regime is started from a single infected node. The application of containment measures to an ongoing epidemic, however, has the effect to make the system enter in its critical regime with a num…
This paper is concerned with the analysis of vaccination strategies in a stochastic SIR (susceptible $\to$ infected $\to$ removed) model for the spread of an epidemic amongst a population of individuals with a random network of social contacts that is also partitioned into households. Under various vaccine action models, we consider both household-based vaccination schemes, in which the way in which individuals are chosen for vaccination depends on the size of the households in which they reside, and acquaintance vaccination, which targets individuals of high degree in the social network. For …
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…
The dynamics of epidemics depend on how people's behavior changes during an outbreak. At the beginning of the epidemic, people do not know about the virus, then, after the outbreak of epidemics and alarm, they begin to comply with the restrictions and the spreading of epidemics may decline. Over time, some people get tired/frustrated by the restrictions and stop following them (exhaustion), especially if the number of new cases drops down. After resting for a while, they can follow the restrictions again. But during this pause the second wave can come and become even stronger then the first on…
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 …
Multiple-type branching processes that model the spread of infectious diseases are investigated. In these stochastic processes, the disease goes through multiple stages before it eventually disappears. We mostly focus on the critical multistage Susceptible-Infected-Recovered (SIR) infection process. In the infinite population limit, we compute the outbreak size distributions and show that asymptotic results apply to more general multiple-type critical branching processes. Finally using heuristic arguments and simulations we establish scaling laws for a multistage SIR model in a finite populati…
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 current outbreak is known as Coronavirus Disease or COVID-19 caused by the virus SAR-COV-2 which continues to wreak havoc across the globe. The World Health Organization (WHO) has declared the outbreak a Public Health Emergency of International Concern. In Pakistan, the spread of the virus is on the rise with the number of infected people and causalities rapidly increasing. In the absence of proper vaccination and treatment, to reduce the number of infections and casualties, the only option so far is to educate people regarding preventive measures and to enforce countrywide lock-down. Any …
As the outbreak of COVID-19 enters its third year, we have now enough data to analyse the behavior of the pandemic with mathematical models over a long period of time. The pandemic alternates periods of high and low infections, in a way that sheds a light on the nature of mathematical model that can be used for reliable predictions. The main hypothesis of the model presented here is that the oscillatory behavior is a structural feature of the outbreak, even without postulating a time-dependence of the coefficients. As such, it should be reflected by the presence of limit cycles as asymptotic s…
We analyze the dynamics of the COVID-19 epidemic taking into account the role of the unreported cases. After a first section in which we deal with a framework of very slow test capacity, we turn to the model recently introduced/implemented by Liu, Magal, Seydi and Webb. First, we prove some basic structural results for the corresponding ODE, as for instance the convergence of S(t) to a positive limit. These are similar to those of the classical SIR model, although the maxima of the corresponding curves are not necessarily unique. Finally, we implement the model -- but with a variable transmiss…
Much effort has been directed towards using mathematical models to understand and predict contagious disease, in particular Ebola outbreaks. Classical SIR (susceptible-infected-recovered) compartmental models capture well the dynamics of the outbreak in certain communities, and accurately describe the differences between them based on a variety of parameters. However, repeated resurgence of Ebola contagions suggests that there are components of the global disease dynamics that we don't yet fully understand and can't effectively control. In order to understand the dynamics of a more widespread …
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…
Contagious respiratory diseases, such as COVID-19, depend on sufficiently prolonged exposures for the successful transmission of the underlying pathogen. It is important for organizations to evaluate the efficacy of interventions aiming at mitigating viral transmission among their personnel. We have developed a operational risk assessment simulation framework that couples a spatial agent-based model of movement with a SIR epidemiological model to assess the relative risks of different intervention strategies. By applying our model on MIT's STATA building, we assess the impacts of three possibl…
Using the classical Susceptible-Infected-Recovered epidemiological model, an analytical formula is derived for the number of beds occupied by Covid-19 patients. The analytical curve is fitted to data in Belgium, France, New York City and Switzerland, with a correlation coefficient exceeding 98.8%, suggesting that finer models are unnecessary with such macroscopic data. The fitting is used to extract estimates of the doubling time in the ascending phase of the epidemic, the mean recovery time and, for those who require medical intervention, the mean hospitalization time. Large variations can be…
The dramatic outbreak of the coronavirus disease 2019 (COVID-19) pandemics and its ongoing progression boosted the scientific community's interest in epidemic modeling and forecasting. The SIR (Susceptible-Infected-Removed) model is a simple mathematical model of epidemic outbreaks, yet for decades it evaded the efforts of the community to derive an explicit solution. The present work demonstrates that this is a non-trivial task. Notably, it is proven that the explicit solution of the model requires the introduction of a new transcendental special function, related to the Wright's Omega functi…
The spread of an epidemic process is considered in the context of a spatial SIR stochastic model that includes a parameter $0\le p\le 1$ that assigns weights $p$ and $1- p$ to global and local infective contacts respectively. The model was previously studied by other authors in different contexts. In this work we characterized the behavior of the system around the threshold for epidemic spreading. We first used a deterministic approximation of the stochastic model and checked the existence of a threshold value of $p$ for exponential epidemic spread. An analytical expression, which defines a fu…
We investigate three aspects of the importance of nodes with respect to Susceptible-Infectious-Removed (SIR) disease dynamics: influence maximization (the expected outbreak size given a set of seed nodes), the effect of vaccination (how much deleting nodes would reduce the expected outbreak size) and sentinel surveillance (how early an outbreak could be detected with sensors at a set of nodes). We calculate the exact expressions of these quantities, as functions of the SIR parameters, for all connected graphs of three to seven nodes. We obtain the smallest graphs where the optimal node sets ar…
The new coronavirus covid-19 has spread very quickly worldwide, leading the World Health Organization (WHO) to declare a state of pandemic. One of the questions many policy makers, and governments are asking themselves is how the spread is going to evolve in time. In this study, I applied an adapted SIR model previously used in South Korea to model the MERS outbreak, which is also caused by a coronavirus, to estimate the evolution of the curve of active cases in the case of the Portuguese situation, using Italian dara. I then construct five different scenarios for the evolution of covid-19 in …
Intent of this research is to explore how mathematical models, specifically Susceptible-Infected-Removed (SIR) model, can be utilized to forecast peak outbreak timeline of COVID-19 epidemic amongst a population of interest starting from the date of first reported case. Till the time of this research, there was no effective and universally accepted vaccine to control transmission and spread of this infection. COVID-19 primarily spreads in population through respiratory droplets from an infected person cough and sneeze which infects people who are in proximity. COVID-19 is spreading contagiously…
COVID-19 pandemic has been raging all around the world for almost a year now, as of November 1, 2020. In this paper, we try to analyze the variation of the COVID-19 pandemic in different countries in the light of some modifications to the susceptible-infected-recovered (SIR) model. The SIR model was modified by taking time-dependent rate parameters. From this modified SIR model, the basic reproduction number, effective reproduction number, herd immunity, and herd immunity threshold are redefined. The re-outbreak of the COVID-19 is a real threat to various countries. We have used the above-ment…
Time varying susceptibility of host at individual level due to waning and boosting immunity is known to induce rich long-term behavior of disease transmission dynamics. Meanwhile, the impact of the time varying heterogeneity of host susceptibility on the shot-term behavior of epidemics is not well-studied, even though the large amount of the available epidemiological data are the short-term epidemics. Here we constructed a parsimonious mathematical model describing the short-term transmission dynamics taking into account natural-boosting immunity by reinfection, and obtained the explicit solut…
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…
Non-pharmaceutical interventions (NPIs) have played a crucial role in controlling the spread of COVID-19. Nevertheless, NPI efficacy varies enormously between and within countries, mainly because of population and behavioural heterogeneity. In this work, we adapted a multi-group SEIRA model to study the spreading dynamics of COVID-19 in Chile, representing geographically separated regions of the country by different groups. We use national mobilization statistics to estimate the connectivity between regions and data from governmental repositories to obtain COVID-19 spreading and death rates in…
The classic SIR model of epidemic dynamics is solved completely by quadratures, including a time integral transform expanded in a series of incomplete gamma functions. The model is also generalized to arbitrary time-dependent infection rates and solved explicitly when the control parameter depends on the accumulated infections at time $t$. Numerical results are presented by way of comparison. Autonomous and non-autonomous generalizations of SIR for interacting regions are also considered, including non-separability for two or more interacting regions. A reduction of simple SIR models to one va…
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 (…
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…
Real-time vaccination following an outbreak can effectively mitigate the damage caused by an infectious disease. However, in many cases, available resources are insufficient to vaccinate the entire at-risk population, logistics result in delayed vaccine deployment, and the interaction between members of different cities facilitates a wide spatial spread of infection. Limited vaccine, time delays, and interaction (or coupling) of cities lead to tradeoffs that impact the overall magnitude of the epidemic. These tradeoffs mandate investigation of optimal strategies that minimize the severity of t…
Transmission rates in epidemic outbreaks may vary over time depending on the societal response. Non-pharmacological mitigation strategies such as social distancing and the adoption of protective equipment aim precisely at reducing transmission rates by reducing infectious contacts. To investigate the effects of mitigation strategies on the evolution of epidemics, nonlinear transmission rates that are influenced by the levels of infections, deaths or recoveries have been included in many variants of the classical SIR model. This class of models is particularly relevant to the COVID-19 epidemic,…
In this paper, we develop a multi-group epidemic framework via virtual dispersal where the risk of infection is a function of the residence time and local environmental risk. This novel approach eliminates the need to define and measure contact rates that are used in the traditional multi-group epidemic models with heterogeneous mixing. We apply this approach to a general $n$-patch SIS model whose basic reproduction number $\mathcal R_0 $ is computed as a function of a patch residence-times matrix $\mathbb P$. Our analysis implies that the resulting $n$-patch SIS model has robust dynamics when…
This paper presents an dashboard developed to analyse the outbreak of the Covid-19 infection in Italy considering daily NUTS-3 data on positive cases provided by the Italian Ministry of Health and on deaths derived from Italian regional authorities' official press conferences. Descriptive time series plots are provided together with a map describing the spatial distribution of province cumulative cases and rates. A section on a proposed time-dependent adjusted SIRD model for NUTS-3 regions is also provided in the dashboard.
When the novel coronavirus disease SARS-CoV2 (COVID-19) was officially declared a pandemic by the WHO in March 2020, the scientific community had already braced up in the effort of making sense of the fast-growing wealth of data gathered by national authorities all over the world. However, despite the diversity of novel theoretical approaches and the comprehensiveness of many widely established models, the official figures that recount the course of the outbreak still sketch a largely elusive and intimidating picture. Here we show unambiguously that the dynamics of the COVID-19 outbreak belong…
Interplay between evolutionary and epidemic time scales challenges the outcome of control policies
University of Zaragoza（萨拉戈萨大学）；Institute for Biocomputation and Physics of Complex Systems (BIFI), University of Zaragoza（萨拉戈萨大学复杂系统生物计算与物理研究所）；Universitat Rovira i Virgili（罗维拉-依维尔吉里大学）；Pacific Northwest National Laboratory（太平洋西北国家实验室）；Complexity Science Hub Vienna（维也纳复杂性科学中心）；ComSCIAM, Universitat Rovira i Virgili（罗维拉-依维尔吉里大学计算科学与数学中心）；Center for Computational Social Science, University of Kobe（神户大学计算社会科学中心）
One of the many unresolved questions that revolves around the Covid-19 pandemic is whether local outbreaks can depend on ambient conditions like temperature and relative humidity. In this paper, we develop a model that tries to explain and describe the temperature and relative humidity sensitivity of respiratory droplets and their possible connection in determining viral outbreaks. The model has two parts. First, we model the growth rate of the infected population based on a reaction mechanism - the final equations of which are similar to the well-known SIR model. The advantage of modeling the…
The current COVID-19 pandemic and subsequent lockdowns have highlighted the close and delicate relationship between a country's public health and economic health. Macroeconomic models that use preexisting epidemic models to calculate the impacts of a disease outbreak are therefore extremely useful for policymakers seeking to evaluate the best course of action in such a crisis. We develop an SIR model of the COVID-19 pandemic that explicitly considers herd immunity, behavior-dependent transmission rates, remote workers, and indirect externalities of lockdown. This model is presented as an exit …
