In the absence of drugs and vaccines, policymakers use non-pharmaceutical interventions such as social distancing to decrease rates of disease-causing contact, with the aim of reducing or delaying the epidemic peak. These measures carry social and economic costs, so societies may be unable to maintain them for more than a short period of time. Intervention policy design often relies on numerical simulations of epidemic models, but comparing policies and assessing their robustness demands clear principles that apply across strategies. Here we derive the theoretically optimal strategy for using a time-limited intervention to reduce the peak prevalence of a novel disease in the classic Susceptible-Infectious-Recovered epidemic model. We show that broad classes of easier-to-implement strategies can perform nearly as well as the theoretically optimal strategy. But neither the optimal strategy nor any of these near-optimal strategies is robust to implementation error: small errors in timing the intervention produce large increases in peak prevalence. Our results reveal fundamental principles of non-pharmaceutical disease control and expose their potential fragility. For robust control, an intervention must be strong, early, and ideally sustained.
Recent evidences show that individuals who recovered from COVID-19 can be reinfected. However, this phenomenon has rarely been studied using mathematical models. In this paper, we propose a SEIRE epidemic model to describe the spread of the epidemic with reinfection. We obtain the important thresholds $R_0$ (the basic reproduction number) and Rc (a threshold less than one). Our investigations show that when $R_0 > 1$, the system has an endemic equilibrium, which is globally asymptotically stable. When $R_c < R_0 < 1$, the epidemic system exhibits bistable dynamics. That is, the system has back…
Compartmental equations are primary tools in disease spreading studies. Their predictions are accurate for large populations but disagree with empirical and simulated data for finite populations, where uncertainties become a relevant factor. Starting from the agent-based approach, we investigate the role of uncertainties and autocorrelation functions in SIS epidemic model, including their relationship with epidemiological variables. We find new differential equations that take uncertainties into account. The findings provide improved predictions to the SIS model and it can offer new insights f…
In this paper, the authors develop a method of detecting correlations between epidemic patterns in different regions that are due to human movement and introduce a null model in which the travel-induced correlations are cancelled. They apply this method to the well-documented cases of seasonal influenza outbreaks in the United States and France. In the United States (using data for 1972-2002), the authors observed strong short-range correlations between several states and their immediate neighbors, as well as robust long-range spreading patterns resulting from large domestic air-traffic flows.…
We present a detailed set-based analysis of the well-known SIR and SEIR epidemic models subjected to hard caps on the proportion of infective individuals, and bounds on the allowable intervention strategies, such as social distancing, quarantining and vaccination. We describe the admissible and maximal robust positively invariant (MRPI) sets of these two models via the theory of barriers. We show how the sets may be used in the management of epidemics, for both perfect and imperfect/uncertain models, detailing how intervention strategies may be specified such that the hard infection cap is nev…
Traditionally, the identification of parameters in the formulation and solution of inverse problems considers that models, variables and mathematical parameters are free of uncertainties. This aspect simplifies the estimation process, but does not consider the influence of relatively small changes in the design variables in terms of the objective function. In this work, the SIDR (Susceptible, Infected, Dead and Recovered) model is used to simulate the dynamic behavior of the novel coronavirus disease (COVID-19), and its parameters are estimated by formulating a robust inverse problem, that is,…
The spread of infectious diseases near the epidemic threshold is investigated. Scaling laws for the size and the duration of outbreaks originating from a single infected individual in a large susceptible population are obtained. The maximal size of an outbreak n_* scales as N^{2/3} with N the population size. This scaling law implies that the average outbreak size <n> scales as N^{1/3}. Moreover, the maximal and the average duration of an outbreak grow as t_* ~ N^{1/3} and <t> ~ ln N, respectively.
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…
An Ebola outbreak of unparalleled size is currently affecting several countries in West Africa, and international efforts to control the outbreak are underway. However, the efficacy of these interventions, and their likely impact on an Ebola epidemic of this size, is unknown. Forecasting and simulation of these interventions may inform public health efforts. We use existing data from Liberia and Sierra Leone to parameterize a mathematical model of Ebola and use this model to forecast the progression of the epidemic, as well as the efficacy of several interventions, including increased contact …
Effective intervention strategies for epidemics rely on the identification of their origin and on the robustness of the predictions made by network disease models. We introduce a Bayesian uncertainty quantification framework to infer model parameters for a disease spreading on a network of communities from limited, noisy observations; the state-of-the-art computational framework compensates for the model complexity by exploiting massively parallel computing architectures. Using noisy, synthetic data, we show the potential of the approach to perform robust model fitting and additionally demonst…
We study the effects of switching social contacts as a strategy to control epidemic outbreaks. Connections between susceptible and infective individuals can be broken by either individual, and then reconnected to a randomly chosen member of the population. It is assumed that the reconnecting individual has no previous information on the epidemiological condition of the new contact. We show that reconnection can completely suppress the disease, both by continuous and discontinuous transitions between the endemic and the infection-free states. For diseases with an asymptomatic phase, we analyze …
Developing robust, quantitative methods to optimize resource allocations in response to epidemics has the potential to save lives and minimize health care costs. In this paper, we develop and apply a computationally efficient algorithm that enables us to calculate the complete probability distribution for the final epidemic size in a stochastic Susceptible-Infected-Recovered (SIR) model. Based on these results, we determine the optimal allocations of a limited quantity of vaccine between two non-interacting populations. We compare the stochastic solution to results obtained for the traditional…
When controlling an emerging outbreak of an infectious disease it is essential to know the key epidemiological parameters, such as the basic reproduction number $R_0$ and the control effort required to prevent a large outbreak. These parameters are estimated from the observed incidence of new cases and information about the infectious contact structures of the population in which the disease spreads. However, the relevant infectious contact structures for new, emerging infections are often unknown or hard to obtain. Here we show that for many common true underlying heterogeneous contact struct…
Contact tracing data collected from disease outbreaks has received relatively little attention in the epidemic modelling literature because it is thought to be unreliable: infection sources might be wrongly attributed, or data might be missing due to resource contraints in the questionnaire exercise. Nevertheless, these data might provide a rich source of information on disease transmission rate. This paper presents novel methodology for combining contact tracing data with rate-based contact network data to improve posterior precision, and therefore predictive accuracy. We present an advanceme…
A simplified susceptible-infected-recovered (SIR) epidemic model and a small-world model are applied to analyse the spread and control of Severe Acute Respiratory Syndrome (SARS) for Hong Kong in early 2003. From data available in mid April 2003, we predict that SARS would be controlled by June and nearly 1700 persons would be infected based on the SIR model. This is consistent with the known data. A simple way to evaluate the development and efficacy of control is described and shown to provide a useful measure for the future evolution of an epidemic. This may contribute to improve strategic …
Tracking dynamics of superspreading through contacts, exposures, and transmissions in edge-based network epidemics
Princeton University（普林斯顿大学）；University of Vermont（佛蒙特大学）；University of Copenhagen（哥本哈根大学）；Roskilde University（罗斯基勒大学）；Emory University（埃默里大学）；Complexity Science Hub（复杂性科学中心）；Santa Fe Institute（圣塔菲研究所）
We study the classic Susceptible-Infected-Recovered (SIR) model for the spread of an infectious disease. In this stochastic process, there are two competing mechanism: infection and recovery. Susceptible individuals may contract the disease from infected individuals, while infected ones recover from the disease at a constant rate and are never infected again. Our focus is the behavior at the epidemic threshold where the rates of the infection and recovery processes balance. In the infinite population limit, we establish analytically scaling rules for the time-dependent distribution functions t…
In this paper, we focus on the influence of heterogeneity and stochasticity of the population on the dynamical structure of a basic susceptible-infected-susceptible (SIS) model. First we prove that, upon a suitable mathematical reformulation of the basic reproduction number, the homogeneous system and the heterogeneous system exhibit a completely analogous global behaviour. Then we consider noise terms to incorporate the fluctuation effects and the random import of the disease into the population and analyse the influence of heterogeneity on warning signs for critical transitions (or tipping p…
Epidemic control is of great importance for human society. Adjusting interacting partners is an effective individualized control strategy. Intuitively, it is done either by shortening the interaction time between susceptible and infected individuals or by increasing the opportunities for contact between susceptible individuals. Here, we provide a comparative study on these two control strategies by establishing an epidemic model with non-uniform stochastic interactions. It seems that the two strategies should be similar, since shortening the interaction time between susceptible and infected in…
Many pathogens spread primarily via direct contact between infected and susceptible hosts. Thus, the patterns of contacts or contact network of a population fundamentally shapes the course of epidemics. While there is a robust and growing theory for the dynamics of single epidemics in networks, we know little about the impacts of network structure on long term epidemic or endemic transmission. For seasonal diseases like influenza, pathogens repeatedly return to populations with complex and changing patterns of susceptibility and immunity acquired through prior infection. Here, we develop two m…
We study epidemic arrival times in meta-population disease models through the lens of front propagation into unstable states. We demonstrate that several features of invasion fronts in the PDE context are also relevant to the network case. We show that the susceptible-infected-recovered model on a network is linearly determined in the sense that the arrival times in the nonlinear system are approximated by the arrival times of the instability in the system linearized near the disease free state. Arrival time predictions are extended to an susceptible-exposed-infected-recovered model. We then s…
In this paper, we propose a new real-time differential virus transmission model, which can give more accurate and robust short-term predictions of COVID-19 transmitted infectious disease with benefits of near-term trend projection. Different from the existing Susceptible-Exposed-Infected-Removed (SEIR) based virus transmission models, which fits well for pandemic modeling with sufficient historical data, the new model, which is also SEIR based, uses short history data to find the trend of the changing disease dynamics for the infected, the dead and the recovered so that it can naturally accomm…
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…
We show that precise knowledge of epidemic transmission parameters is not required to build an informative model of the spread of disease. We propose a detailed model of the topology of the contact network under various external control regimes and demonstrate that this is sufficient to capture the salient dynamical characteristics and to inform decisions. Contact between individuals in the community is characterised by a contact graph, the structure of that contact graph is selected to mimic community control measures. Our model of city-level transmission of an infectious agent (SEIR model) c…
Market trade-routes can support infectious-disease transmission, impacting biological populations and even disrupting causal trade. Epidemiological models increasingly account for reductions in infectious contact, such as risk-aversion behaviour in response to pathogen outbreaks. However, market dynamics clearly differ from simple risk-aversion, as are driven by different motivation and conditioned by trade constraints, known in economics as friction, that arise because exchanges are costly. Here we develop a novel economic-market model where transient and long-term market dynamics are determi…
The success of a vaccination program is crucially dependent on its adoption by a critical fraction of the population, as the resulting herd immunity prevents future outbreaks of an epidemic. However, the effectiveness of a campaign can engender its own undoing if individuals choose to not get vaccinated in the belief that they are protected by herd immunity. Although this may appear to be an optimal decision, based on a rational appraisal of cost and benefits to the individual, it exposes the population to subsequent outbreaks. We investigate if voluntary vaccination can emerge in a an integra…
A probabilistic approach to the epidemic evolution on realistic social-contact networks allows for characteristic differences among subjects, including the individual number and structure of social contacts, and the heterogeneity of the infection and recovery rates according to age or medical preconditions. Within our probabilistic Susceptible-Infectious-Removed (SIR) model on social-contact networks, we evaluate the `infection load' or `activation margin' of various control scenarios; by confinement, by vaccination, and by their combination. We compare the epidemic burden for subpopulations w…
The transmission of vector infectious diseases, which produces complex spatiotemporal patterns, is analyzed by a periodically forced two-dimensional cellular automata model. The system, which comprises three population levels, is introduced to describe complex features of the dynamics of the vector transmitted dengue epidemics, known to be very sensitive to seasonal variables. The three coupled levels represent the human, the adult and immature vector populations. The dynamics includes external seasonality forcing (rainfall intensity data), human and mosquito mobility, and vector control effec…
Coronavirus COVID-19 spreads through the population mostly based on social contact. To gauge the potential for widespread contagion, to cope with associated uncertainty and to inform its mitigation, more accurate and robust modelling is centrally important for policy making. We provide a flexible modelling approach that increases the accuracy with which insights can be made. We use this to analyse different scenarios relevant to the COVID-19 situation in the UK. We present a stochastic model that captures the inherently probabilistic nature of contagion between population members. The computat…
The COVID-19 epidemic has been spreading around the world for nearly three years, and asymptomatic infections have exacerbated the spread of the epidemic. To evaluate the role of asymptomatic infections in the spread of the epidemic, we develop mathematical models to assess the proportion of asymptomatic infections caused by different strains of the main covid-19 variants. The analysis shows that when the control reproduction number is less than 1, the disease-free equilibrium point of the model is globally asymptotically stable; and when the control reproduction number is greater than 1, the …
