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Register a new userExploring the threshold of epidemic spreading for a stochastic SIR model with local and global contacts
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The spread of an epidemic process is considered in the context of a spatial SIR stochastic model that includes a parameter $0le ple 1$ 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 function of the quotient $alpha$ between the transmission and recovery rates, is obtained to approximate this threshold. We then performed different analyses based on intensive stochastic simulations and found that this expression is also a good estimate for a similar threshold value of $p$ obtained in the stochastic model. The dynamics of the average number of infected individuals and the average size of outbreaks show a behavior across the threshold that is well described by the deterministic approximation. The distributions of the outbreak sizes at the threshold present common features for all the cases considered corresponding to different values of $alpha>1$. These features are otherwise already known to hold for the standard stochastic SIR model at its threshold, $alpha=1$: (i) the probability of having an outbreak of size $n$ goes asymptotically as $n^{-3/2}$ for an infinite system, (ii) the maximal size of an outbreak scales as $N^{2/3}$ for a finite system of size $N$.
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We study a simple realistic model for describing the diffusion of an infectious disease on a population of individuals. The dynamics is governed by a single functional delay differential equation, which, in the case of a large population, can be solved exactly, even in the presence of a time-dependent infection rate. This delay model has a higher degree of accuracy than that of the so-called SIR model, commonly used in epidemiology, which, instead, is formulated in terms of ordinary differential equations. We apply this model to describe the outbreak of the new infectious disease, Covid-19, in Italy, taking into account the containment measures implemented by the government in order to mitigate the spreading of the virus and the social costs for the population.
We study the stochastic susceptible-infected-recovered (SIR) model with time-dependent forcing using analytic techniques which allow us to disentangle the interaction of stochasticity and external forcing. The model is formulated as a continuous time Markov process, which is decomposed into a deterministic dynamics together with stochastic corrections, by using an expansion in inverse system size. The forcing induces a limit cycle in the deterministic dynamics, and a complete analysis of the fluctuations about this time-dependent solution is given. This analysis is applied when the limit cycle is annual, and after a period-doubling when it is biennial. The comprehensive nature of our approach allows us to give a coherent picture of the dynamics which unifies past work, but which also provides a systematic method for predicting the periods of oscillations seen in whooping cough and measles epidemics.
Pairwise models are used widely to model epidemic spread on networks. These include the modelling of susceptible-infected-removed (SIR) epidemics on regular networks and extensions to SIS dynamics and contact tracing on more exotic networks exhibiting degree heterogeneity, directed and/or weighted links and clustering. However, extra features of the disease dynamics or of the network lead to an increase in system size and analytical tractability becomes problematic. Various `closures can be used to keep the system tractable. Focusing on SIR epidemics on regular but clustered networks, we show that even for the most complex closure we can determine the epidemic threshold as an asymptotic expansion in terms of the clustering coefficient.We do this by exploiting the presence of a system of fast variables, specified by the correlation structure of the epidemic, whose steady state determines the epidemic threshold. While we do not find the steady state analytically, we create an elegant asymptotic expansion of it. We validate this new threshold by comparing it to the numerical solution of the full system and find excellent agreement over a wide range of values of the clustering coefficient, transmission rate and average degree of the network. The technique carries over to pairwise models with other closures [1] and we note that the epidemic threshold will be model dependent. This emphasises the importance of model choice when dealing with realistic outbreaks.
We consider an SIR-type (Susceptible $to$ Infected $to$ Recovered) stochastic epidemic process with multiple modes of transmission on a contact network. The network is given by a random graph following a multilayer configuration model where edges in different layers correspond to potentially infectious contacts of different types. We assume that the graph structure evolves in response to the epidemic via activation or deactivation of edges. We derive a large graph limit theorem that gives a system of ordinary differential equations (ODEs) describing the evolution of quantities of interest, such as the proportions of infected and susceptible vertices, as the number of nodes tends to infinity. Analysis of the limiting system elucidates how the coupling of edge activation and deactivation to infection status affects disease dynamics, as illustrated by a two-layer network example with edge types corresponding to community and healthcare contacts. Our theorem extends some earlier results deriving the deterministic limit of stochastic SIR processes on static, single-layer configuration model graphs. We also describe precisely the conditions for equivalence between our limiting ODEs and the systems obtained via pair approximation, which are widely used in the epidemiological and ecological literature to approximate disease dynamics on networks. Potential applications include modeling Ebola dynamics in West Africa, which was the motivation for this study.
We propose a simple SIR model in order to investigate the impact of various confinement strategies on a most virulent epidemic. Our approach is motivated by the current COVID-19 pandemic. The main hypothesis is the existence of two populations of susceptible persons, one which obeys confinement and for which the infection rate does not exceed 1, and a population which, being non confined for various imperatives, can be substantially more infective. The model, initially formulated as a differential system, is discretised following a specific procedure, the discrete system serving as an integrator for the differential one. Our model is calibrated so as to correspond to what is observed in the COVID-19 epidemic. Several conclusions can be reached, despite the very simple structure of our model. First, it is not possible to pinpoint the genesis of the epidemic by just analysing data from when the epidemic is in full swing. It may well turn out that the epidemic has reached a sizeable part of the world months before it became noticeable. Concerning the confinement scenarios, a universal feature of all our simulations is that relaxing the lockdown constraints leads to a rekindling of the epidemic. Thus we sought the conditions for the second epidemic peak to be lower than the first one. This is possible in all the scenarios considered (abrupt, progressive or stepwise exit) but typically a progressive exit can start earlier than an abrupt one. However, by the time the progressive exit is complete, the overall confinement times are not too different. From our results, the most promising strategy is that of a stepwise exit. And in fact its implementation could be quite feasible, with the major part of the population (minus the fragile groups) exiting simultaneously but obeying rigorous distancing constraints.
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