Models of biological processes are often subject to different sources of noise. Developing an understanding of the combined effects of different types of uncertainty is an open challenge. In this paper, we study a variant of the susceptible-infective-recovered model of epidemic spread, which combines both agent-to-agent heterogeneity and intrinsic noise. We focus on epidemic cycles, driven by the stochasticity of infection and recovery events, and study in detail how heterogeneity in susceptibilities and propensities to pass on the disease affects these quasi-cycles. While the system can only be described by a large hierarchical set of equations in the transient regime, we derive a reduced closed set of equations for population-level quantities in the stationary regime. We analytically obtain the spectra of quasi-cycles in the linear-noise approximation. We find that the characteristic frequency of these cycles is typically determined by population averages of susceptibilities and infectivities, but that their amplitude depends on higher-order moments of the heterogeneity. We also investigate the synchronisation properties and phase lag between different groups of susceptible and infected individuals.
In this paper, we show how to modify a compartmental epidemic model, without changing the dimension, such that separable static heterogeneity is taken into account. The derivation is based on the Kermack-McKendrick renewal equation.
Surveillance data serving for epidemic alert systems are typically fully aggregated in space. However, epidemics may be spatially heterogeneous, undergoing distinct dynamics in distinct regions of the surveillance area. We unveil this in retrospective analyses by classifying incidence time series. We use Pearson correlation to quantify the similarity between local time series and then classify them using modularity maximization. The surveillance area is thus divided into regions with different incidence patterns. We analyzed 31 years of data on influenza-like-illness from the French system Sen…
We generalize the Susceptible-Infected-Removed model for epidemics to take into account generic effects of heterogeneity in the degree of susceptibility to infection in the population. We introduce a single new parameter corresponding to a power-law exponent of the susceptibility distribution that characterizes the population heterogeneity. We show that our generalized model is as simple as the original model which is contained as a limiting case. Because of this simplicity, numerical solutions can be generated easily and key properties of the epidemic wave can still be obtained exactly. In pa…
Population heterogeneity, especially in individuals' contact networks, plays an important role in transmission dynamics of infectious diseases. For vaccine-preventable diseases, outstanding issues like vaccine hesitancy and availability of vaccines further lead to nonuniform coverage among groups, not to mention the efficacy of vaccines and the mixing pattern varying from one group to another. As the ongoing COVID-19 pandemic transitions to endemicity, it is of interest and significance to understand the impact of aforementioned population heterogeneity on the emergence and persistence of epid…
The impact of heterogeneity on the co-evolution of cooperation and epidemic spreading in complex networks
Institute for Advanced Studies in Basic Sciences (IASBS)（基础科学高等研究院）；Max Planck Institute of Animal Behavior（马克斯·普朗克动物行为研究所）；University of Konstanz（康斯坦茨大学）
Epidemics generally spread through a succession of waves that reflect factors on multiple timescales. On short timescales, super-spreading events lead to burstiness and overdispersion, while long-term persistent heterogeneity in susceptibility is expected to lead to a reduction in the infection peak and the herd immunity threshold (HIT). Here, we develop a general approach to encompass both timescales, including time variations in individual social activity, and demonstrate how to incorporate them phenomenologically into a wide class of epidemiological models through parameterization. We deriv…
We study the impact of contact heterogeneity on epidemic dynamics. A system characterized by multiple susceptible populations is considered. The description of the spread of an infectious disease is obtained through the study of a system of Boltzmann-type equations for the number densities of social contacts of the introduced compartments. A macroscopic system of equations characterizing observable effects of the epidemic is then derived to assess the impact of contact heterogeneity.
The basic reproductive number -- $R_0$ -- is one of the most common and most commonly misapplied numbers in public health. Although often used to compare outbreaks and forecast pandemic risk, this single number belies the complexity that two different pathogens can exhibit, even when they have the same $R_0$. Here, we show how to predict outbreak size using estimates of the distribution of secondary infections, leveraging both its average $R_0$ and the underlying heterogeneity. To do so, we reformulate and extend a classic result from random network theory that relies on contact tracing data t…
The spread of COVID-19 has been thwarted in most countries through non-pharmaceutical interventions. In particular, the most effective measures in this direction have been the stay-at-home and closure strategies of businesses and schools. However, population-wide lockdowns are far from being optimal carrying heavy economic consequences. Therefore, there is nowadays a strong interest in designing more efficient restrictions. In this work, starting from a recent kinetic-type model which takes into account the heterogeneity described by the social contact of individuals, we analyze the effects of…
Epidemic disease spreading is conventionally often modelled and analyzed by means of rate and diffusion equations, following the paradigms of well-controlled chemical reactions and diffusive dynamics in a test tube. Yet, serious worries that this suggestive and appealing similarity might be a false friend were already voiced by the pioneers of mathematical epidemiology. A century later, we can draw on cross-fertilizations from network and game theory and the emerging field of eco-evolutionary dynamics to substantiate them. Epidemiological spreading is thereby revealed as a fundamentally hetero…
Most models of epidemic spread, including many designed specifically for COVID-19, implicitly assume mass-action contact patterns and undirected contact networks, meaning that the individuals most likely to spread the disease are also the most at risk to receive it from others. Here, we review results from the theory of random directed graphs which show that many important quantities, including the reproduction number and the epidemic size, depend sensitively on the joint distribution of in- and out-degrees ("risk" and "spread"), including their heterogeneity and the correlation between them. …
We introduce a mathematical description of the impact of sociality in the spread of infectious diseases by integrating an epidemiological dynamics with a kinetic modeling of population-based contacts. The kinetic description leads to study the evolution over time of Boltzmann-type equations describing the number densities of social contacts of susceptible, infected and recovered individuals, whose proportions are driven by a classical SIR-type compartmental model in epidemiology. Explicit calculations show that the spread of the disease is closely related to moments of the contact distribution…
We analyse an extended SIRS epidemic model in which immunity at the individual level wanes gradually at exponential rate, but where the waning rate may differ between individuals, for instance as an effect of differences in immune systems. The model also includes vaccination schemes aimed to reach and maintain herd immunity. We consider both the informed situation where the individual waning parameters are known, thus allowing selection of vaccinees being based on both time since last vaccination as well as on the individual waning rate, and the more likely uninformed situation where individua…
We model further development of the COVID-19 epidemic in the UK given the current data and assuming different scenarios of handling the epidemic. In this research, we further extend the stochastic model suggested in \cite{us} and incorporate in it all available to us knowledge about parameters characterising the behaviour of the virus and the illness induced by it. The models we use are flexible, comprehensive, fast to run and allow us to incorporate the following: -time-dependent strategies of handling the epidemic; -spatial heterogeneity of the population and heterogeneity of development of …
The interplay of biological, social, structural and random factors makes disease forecasting extraordinarily complex. The course of an epidemic exhibits average growth dynamics determined by features of the pathogen and the population, yet also features significant variability reflecting the stochastic nature of disease spread. In this work, we reframe a stochastic branching process analysis in terms of probability generating functions and compare it to continuous time epidemic simulations on networks. In doing so, we predict the diversity of emerging epidemic courses on both homogeneous and h…
Assessing the performance of compartmental and renewal models for learning $R_{t}$ using spatially heterogeneous epidemic simulations on real geographies
It is well recognized that population heterogeneity plays an important role in the spread of epidemics. While individual variations in social activity are often assumed to be persistent, i.e. constant in time, here we discuss the consequences of dynamic heterogeneity. By integrating the stochastic dynamics of social activity into traditional epidemiological models we demonstrate the emergence of a new long timescale governing the epidemic in broad agreement with empirical data. Our model captures multiple features of real-life epidemics such as COVID-19, including prolonged plateaus and multip…
Spectral analysis characterises oscillatory time series behaviours such as cycles, but accurate estimation requires reasonable numbers of observations. Current COVID-19 time series for many countries are short: pre- and post-lockdown series are shorter still. Accurate estimation of potentially interesting cycles within such series seems beyond reach. We solve the problem of obtaining accurate estimates from short time series by using recent Bayesian spectral fusion methods. Here we show that transformed new daily COVID-19 cases for many countries generally contain three cycles operating at wav…
The correct evaluation of the reproductive number $R$ for COVID-19 -- which characterizes the average number of secondary cases generated by each typical primary case -- is central in the quantification of the potential scope of the pandemic and the selection of an appropriate course of action. In most models, $R$ is modeled as a universal constant for the virus across outbreak clusters and individuals -- effectively averaging out the inherent variability of the transmission process due to varying individual contact rates, population densities, demographics, or temporal factors amongst many. Y…
Many countries are managing COVID-19 epidemic by switching between lighter and heavier restrictions. While an open-close and a close-open cycle have comparable socio-economic costs, the former leads to a much heavier burden in terms of deaths and pressure on the healthcare system. An empirical demonstration of the toll ensuing from procrastination was recently observed in Israel, where both cycles were enforced from late August to mid-December 2020, yielding some 1,600 deaths with open-close compared to 440 with close-open.
Raw data on the cumulative number of deaths at a country level generally indicate a spatially variable distribution of the incidence of COVID-19 disease. An important issue is to determine whether this spatial pattern is a consequence of environmental heterogeneities, such as the climatic conditions, during the course of the outbreak. Another fundamental issue is to understand the spatial spreading of COVID-19. To address these questions, we consider four candidate epidemiological models with varying complexity in terms of initial conditions, contact rates and non-local transmissions, and we f…
We propose a theoretical framework for the study of epidemics in structured metapopulations, with heterogeneous agents, subjected to recurrent mobility patterns. We propose to represent the heterogeneity in the composition of the metapopulations as layers in a multiplex network, where nodes would correspond to geographical areas and layers account for the mobility patterns of agents of the same class. We analyze both the classical Susceptible-Infected-Susceptible and the Susceptible-Infected-Removed epidemic models within this framework, and compare macroscopic and microscopic indicators of th…
Stochastic lattice gas model describing the dynamics of an epidemic
我们研究了一个描述流行病在由三类个体组成的群体中传播动力学开始的随机过程：易感者（S），感染者（I）和康复者（R）。随机过程由局部规则定义，并涉及以下循环过程：S $\to$ I $\to$ R $\to$ S（SIRS）。作为SIRS过程的一个特例，研究了开过程S $\to$ I $\to$ R（SIR）.流行病的过程进行了分析，在不同层次的描述：随机格子气模型和出生和死亡过程。通过蒙特卡罗模拟和动力学平均场近似，我们证明了SIRS随机格子气模型存在一条临界点线，该临界点线将两个相分离：一个是充满S个体的吸收相，另一个是S、I和R个体共存的活跃相，这可能存在也可能不存在种群循环。临界线，这对应于传染病传播的发病，是属于有向渗流普适类。通过考虑生灭过程，我们分析了噪声在稳定振荡中的作用。
Nowcasting and forecasting of epidemic spreading rely on incidence series of reported cases to derive the fundamental epidemiological parameters for a given pathogen. Two relevant drawbacks for predictions are the unknown fractions of undocumented cases and levels of nonpharmacological interventions, which span highly heterogeneously across different places and times. We describe a simple data-driven approach using a compartmental model including asymptomatic and presymptomatic contagions that allows to estimate both the level of undocumented infections and the value of effective reproductive …
