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, randomization of the sequence of the events increases the mean outbreak size of high risk cases. On the other hand, these correlations don't have a consistent effect on the independent infection dynamics, and can either decrease or increase this mean. While randomization of the daily pattern correlations has no significant effect on the size of outbreak in either of the coinfection or independent spreading cases. We also observer that an increase in the mean outbreak size doesn't always coincide with an increase in the outbreak probability; therefore we argue that merely considering the mean outbreak size of all realizations may lead us into misestimating the outbreak risks. Our results suggest that some sort of randomizing contacts in organization level of schools, events or hospitals might help to suppress the spreading dynamics while the risk of an outbreak is high.
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 res…
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 …
We study the spreading of cooperative infections in an empirical temporal network of contacts between people, including health care workers and patients, in a hospital. The system exhibits a phase transition leading to one or several endemic branches, depending on the connectivity pattern and the temporal correlations. There are two endemic branches in the original setting and the non-cooperative case. However, the cooperative interaction between infections reinforces the upper branch, leading to a smaller epidemic threshold and a higher probability for having a big outbreak. We show the micro…
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…
In this paper, we introduce a general framework for co-infection as cooperative SIR dynamics. We first solve analytically CGCG model [1] and then the generalized model in symmetric scenarios. We calculate transition points, order parameter, i.e. total number of infected hosts. Also we show analytically there is a saddle-node bifurcation for two cooperative SIR dynamics and the transition is hybrid. Moreover, we investigate where symmetric solution is stable for initial fluctuations. Then we study asymmetric cases of parameters. The more asymmetry, for the primary and secondary infection rates …
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…
Motivated by recent epidemic outbreaks, including those of COVID-19, we solve the canonical problem of calculating the dynamics and likelihood of extensive outbreaks in a population within a large class of stochastic epidemic models with demographic noise, including the Susceptible-Infected-Recovered (SIR) model and its general extensions. In the limit of large populations, we compute the probability distribution for all extensive outbreaks, including those that entail unusually large or small (extreme) proportions of the population infected. Our approach reveals that, unlike other well-known …
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…
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 …
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 …
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…
We calculate both the exponential and pre-factor contributions in a WKB approximation of the master equation for a stochastic SIR model with highly oscillatory dynamics. Fixing the basic parameters of the model we investigate how the outbreak distribution changes with the population size. We show that this distribution rapidly becomes highly non-Gaussian, acquiring large tails indicating the presence of rare, but large outbreaks, as the population is made smaller. The analytic results are found to be in excellent agreement with simulations until the systems become so small that the dynamics ar…
Since the end of 2019 an outbreak of a new strain of coronavirus, called 2019--nCoV, is reported from China and later other parts of the world. Since January 21, WHO reports daily data on confirmed cases and deaths from both China and other countries. In this work we present some discrete and continuous models to discribe the disease dynamics in China and estimate the needed epidemiological parameters. Good agreement with the current dynamics has be found for both a discrete transmission model and a slightly modified SIR-model.
Contact patterns in populations fundamentally influence the spread of infectious diseases. Current mathematical methods for epidemiological forecasting on networks largely assume that contacts between individuals are fixed, at least for the duration of an outbreak. In reality, contact patterns may be quite fluid, with individuals frequently making and breaking social or sexual relationships. Here we develop a mathematical approach to predicting disease transmission on dynamic networks in which each individual has a characteristic behavior (typical contact number), but the identities of their c…
We have designed a computational model of a virus spread near the outbreak threshold. Using computer simulation we studied the Susceptible - Infected - Recovered (SIR) process where in consequence of a force of habit that is manifested by the population mobility patterns, the recovered persons create the spatio-temporal patterns as the barriers to a virus transmission. The results show a spontaneous stopping of the virus spread without a need to infect the whole population, a non-trivial random noise of daily count of infected cases, and power laws of a cumulative count of infected cases. Outb…
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…
Initial hopes of quickly eradicating the COVID-19 pandemic proved futile, and the goal shifted to controlling the peak of the infection, so as to minimize the load on healthcare systems. To that end, public health authorities intervened aggressively to institute social distancing, lock-down policies, and other Non-Pharmaceutical Interventions (NPIs). Given the high social, educational, psychological, and economic costs of NPIs, authorities tune them, alternatively tightening up or relaxing rules, with the result that, in effect, a relatively flat infection rate results. For example, during the…
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…
Coinfection is the process by which a host that is infected with a pathogen becomes infected by a second pathogen at a later point in time. An immunosuppressant host response to a primary disease can facilitate spreading of a subsequent emergent pathogen among the population. Social contact patterns within the substrate populace can be modelled using complex networks and it has been shown that contact patterns vastly influence the emergent disease dynamics. In this paper, we consider the effect of contact clustering on the coinfection dynamics of two pathogens spreading over a network. We use …
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）
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…
Investigating the effect of adaptive optimal control function in epidemic dynamics: predictions and strategy evolution based on SIR/V game theoretic framework
We are interested in a variation of the SIR (Susceptible/Infected/Recovered) dynamics on the complete graph, in which infected individuals may only spread to neighboring susceptible individuals at fixed rate $λ>0$ while recovered individuals may only spread to neighboring infected individuals at fixed rate $1$. This is also a variant of the so-called chase-escape process introduced by Kordzakhia and then Bordenave & Ganesan. Our work is the first study of this dynamics on complete graphs. Starting with one infected and one recovered individuals on the complete graph with $N+2$ vertices, and st…
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 …
Knowing which individuals can be more efficient in spreading a pathogen throughout a determinate environment is a fundamental question in disease control. Indeed, over the last years the spread of epidemic diseases and its relationship with the topology of the involved system have been a recurrent topic in complex network theory, taking into account both network models and real-world data. In this paper we explore possible correlations between the heterogeneous spread of an epidemic disease governed by the susceptible-infected-recovered (SIR) model, and several attributes of the originating ve…
Random networks with specified degree distributions have been proposed as realistic models of population structure, yet the problem of dynamically modeling SIR-type epidemics in random networks remains complex. I resolve this dilemma by showing how the SIR dynamics can be modeled with a system of three nonlinear ODE's. The method makes use of the probability generating function (PGF) formalism for representing the degree distribution of a random network and makes use of network-centric quantities such as the number of edges in a well-defined category rather than node-centric quantities such as…
The evolution of RNA viruses such as HIV, Hepatitis C and Influenza virus occurs so rapidly that the viruses' genomes contain information on past ecological dynamics. Hence, we develop a phylodynamic method that enables the joint estimation of epidemiological parameters and phylogenetic history. Based on a compartmental susceptible-infected-removed (SIR) model, this method provides separate information on incidence and prevalence of infections. Detailed information on the interaction of host population dynamics and evolutionary history can inform decisions on how to contain or entirely avoid d…
