Rose Baker
A model of reactive social distancing in epidemics is proposed, in which the infection rate changes with the number infected. The final-size equation for the total number that the epidemic will infect can be derived analytically, as can the peak infection proportion. This model could assist planners during for example the COVID-19 pandemic.
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The SARS-CoV-2 infectious outbreak has rapidly spread across the globe and precipitated varying policies to effectuate physical distancing to ameliorate its impact. In this study, we propose a new hybrid machine learning model, SIRNet, for forecasting the spread of the COVID-19 pandemic that couples with the epidemiological models. We use categorized spatiotemporally explicit cellphone mobility data as surrogate markers for physical distancing, along with population weighted density and other local data points. We demonstrate at varying geographical granularity that the spectrum of physical di…
To describe the dynamics of social distancing during pandemics, we follow previous efforts to combine basic epidemiology models (e.g. SIR - Susceptible, Infected, and Recovered) with game and economy theory tools. We present an extension of the SIR model that predicts a series of discontinuous transitions in social distancing. Each transition resembles a phase transition of the second-order (Ginzburg-Landau instability) and, therefore, potentially a general phenomenon. The first wave of COVID-19 led to social distancing around the globe: severe lockdowns to stop the pandemic were followed by a…
The 1918 influenza pandemic was characterized by multiple epidemic waves. We investigated into reactive social distancing, a form of behavioral responses, and its effect on the multiple influenza waves in the United Kingdom. Two forms of reactive social distancing have been used in previous studies: Power function, which is a function of the proportion of recent influenza mortality in a population, and Hill function, which is a function of the actual number of recent influenza mortality. Using a simple epidemic model with a Power function and one common set of parameters, we provided a good mo…
Different countries -- and sometimes different regions within the same countries -- have adopted different strategies in trying to contain the ongoing COVID-19 epidemic; these mix in variable parts social confinement, early detection and contact tracing. In this paper we discuss the different effects of these ingredients on the epidemic dynamics; the discussion is conducted with the help of two simple models, i.e. the classical SIR model and the recently introduced variant (A-SIR, arXiv:2003.08720) which takes into account the presence of a large set of asymptomatic infectives.
This document aims to estimate and describe the effects of the social distancing measures implemented in several countries with the expectancy of controlling the spread of COVID-19. The procedure relies on the classic Susceptible-Infected-Removed (SIR) model, which is modified to incorporate a permissiveness index, representing the isolation achieved by the social distancing and the future development of vaccination campaigns and allowing the math model to reproduce more than one infection wave. The adjusted SIR models are used to study the compromise between the economy's reactivation and the…
The current COVID-19 pandemic has proven that proper control and prevention of infectious disease require creating and enforcing the appropriate public policies. One critical policy imposed by the policymakers is encouraging the population to practice social distancing (i.e. controlling the contact rate among the population). Here we pose a mean-field game model of individuals each choosing a dynamic strategy of making contacts, given the trade-off of gaining utility but also risking infection from additional contacts. We compute and compare the mean-field equilibrium (MFE) strategy, which ass…
We present a modified age-structured SIR model based on known patterns of social contact and distancing measures within Washington, USA. We find that population age-distribution has a significant effect on disease spread and mortality rate, and contribute to the efficacy of age-specific contact and treatment measures. We consider the effect of relaxing restrictions across less vulnerable age-brackets, comparing results across selected groups of varying population parameters. Moreover, we analyze the mitigating effects of vaccinations and examine the effectiveness of age-targeted distributions.…
The most important features to assess the severity of an epidemic are its size and its timescale. We discuss these features in a systematic way in the context of SIR and SIR-type models. We investigate in detail how the size and timescale of the epidemic can be changed by acting on the parameters characterizing the model. Using these results and having as guideline the COVID-19 epidemic in Italy, we compare the efficiency of different containment strategies for contrasting an epidemic diffusion such as social distancing, lockdown, tracing, early detection and isolation.
We analyze an optimal control version of a simple SIR epidemiology model that includes a partially specified vaccination policy and takes into account fatigue from protracted application of social distancing measures. The model assumes demographic (age related) categories, and is otherwise homogeneous. Maximum capacity for vaccination is exogenous and the authorities select its allocation to the categories. They can also adopt measures to uniformly diminish the contact rate between infected and susceptible individuals at an economic cost. Total or partial immunity is modeled, while the contact…
We present an empirical algorithm to forecast the evolution of the number of COVID-19 symptomatic patients in the early stages of the pandemic spread and after strict social distancing interventions. The algorithm is based on a low dimensional model for the variation of the exponential growth rate that decreases after the implementation of strict social distancing measures. From the observable data given by the number of tested positive, our model estimates the number of infected hindcast introducing in the model formulation the incubation time. We also use the model to follow the number of in…
Social distancing, often in the form of lockdown, has been adopted by many countries as a way to contrast the spreading of COVID-19. We discuss the temporal aspects of social distancing in contrasting an epidemic diffusion. We argue that a strategy based uniquely on social distancing requires to maintain the relative measures for a very long time, while a more articulate strategy, which also uses early detection and prompt isolation, can be both more efficient on reducing the epidemic peak and allow to relax the social distancing measures after a much shorter time. We consider in more detail t…
Until a vaccine or therapy is found against the SARS-CoV-2 coronavirus, reaching herd immunity appears to be the only mid-term option. However, if the number of infected individuals decreases and eventually fades only beyond this threshold, a significant proportion of susceptible may still be infected until the epidemic is over. A containment strategy is likely the best policy in the worst case where no vaccine or therapy is found. In order to keep the number of newly infected persons to a minimum, a possible strategy is to apply strict containment measures, so that the number of susceptible i…
How does social distancing affect the reach of an epidemic in social networks? We present Monte Carlo simulation results of a capacity constrained Susceptible-Infected-Removed (SIR) model. The key modelling feature is that individuals are limited in the number of acquaintances that they can interact with, thereby constraining disease transmission to an infectious subnetwork of the original social network. While increased social distancing always reduces the spread of an infectious disease, the magnitude varies greatly depending on the topology of the network. Our results also reveal the import…
For preventing the spread of epidemics such as the coronavirus disease COVID-19, social distancing and the isolation of infected persons are crucial. However, existing reaction-diffusion equations for epidemic spreading are incapable of describing these effects. We present an extended model for disease spread based on combining an SIR model with a dynamical density functional theory where social distancing and isolation of infected persons are explicitly taken into account. The model shows interesting nonequilibrium phase separation associated with a reduction of the number of infections, and …
We study a multi-type SIR epidemic process among a heterogeneous population that interacts through a network. When we base social contact on a random graph with given vertex degrees, we give limit theorems on the fraction of infected individuals. For a given social distancing individual strategies, we establish the epidemic reproduction number $R_0$ which can be used to identify network vulnerability and inform vaccination policies. In the second part of the paper we study the equilibrium of the social distancing game, in which individuals choose their social distancing level according to an a…
We demonstrate that the epidemic renormalisation group approach to pandemics provides an effective and simple way to investigate the dynamics of disease transmission and spreading across different regions of the world. The framework also allows for reliable projections on the impact of travel limitations and social distancing measures on global epidemic spread. We test and calibrate it on reported cases while unveiling the mechanism that governs the delay in the relative peaks of newly infected cases among different regions of the globe. We discover that social distancing measures are more eff…
The spread of COVID-19 during the initial phase of the first half of 2020 was curtailed to a larger or lesser extent through measures of social distancing imposed by most countries. In this work, we link directly, through machine learning techniques, infection data at a country level to a single number that signifies social distancing effectiveness. We assume that the standard SIR model gives a reasonable description of the dynamics of spreading, and thus the social distancing aspect can be modeled through time-dependent infection rates that are imposed externally. We use an exponential ansatz…
COVID-19 continues to spread across the country and around the world. Current strategies for managing the spread of COVID-19 include social distancing. We present VERA, an interactive AI tool, that first enables users to specify conceptual models of the impact of social distancing on the spread of COVID-19. Then, VERA automatically spawns agent-based simulations from the conceptual models, and, given a data set, automatically fills in the values of the simulation parameters from the data. Next, the user can view the simulation results, and, if needed, revise the simulation parameters and run a…
In this paper, we formulate a Nonlinear Model Predictive Control (NMPC) to plan appropriate social distancing measures (and relaxations) in order to mitigate the COVID-19 pandemic effects, considering the contagion development in Brazil. The NMPC strategy is designed upon an adapted data-driven Susceptible-Infected-Recovered-Deceased (SIRD) contagion model, which takes into account the effects of social distancing. Furthermore, the adapted SIRD model includes time-varying auto-regressive contagion parameters, which dynamically converge according to the stage of the pandemic. This new model is …
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,…
I estimate the Susceptible-Infected-Recovered (SIR) epidemic model for Coronavirus Disease 2019 (COVID-19). The transmission rate is heterogeneous across countries and far exceeds the recovery rate, which enables a fast spread. In the benchmark model, 28% of the population may be simultaneously infected at the peak, potentially overwhelming the healthcare system. The peak reduces to 6.2% under the optimal mitigation policy that controls the timing and intensity of social distancing. A stylized asset pricing model suggests that the stock price temporarily decreases by 50% in the benchmark case …
The outbreak of the novel coronavirus, COVID-19, has been declared a pandemic by the WHO. The structures of social contact critically determine the spread of the infection and, in the absence of vaccines, the control of these structures through large-scale social distancing measures appears to be the most effective means of mitigation. Here we use an age-structured SIR model with social contact matrices obtained from surveys and Bayesian imputation to study the progress of the COVID-19 epidemic in India. The basic reproductive ratio R0 and its time-dependent generalization are computed based o…
We present three data driven model-types for COVID-19 with a minimal number of parameters to provide insights into the spread of the disease that may be used for developing policy responses. The first is exponential growth, widely studied in analysis of early-time data. The second is a self-exciting branching process model which includes a delay in transmission and recovery. It allows for meaningful fit to early time stochastic data. The third is the well-known Susceptible-Infected-Resistant (SIR) model and its cousin, SEIR, with an "Exposed" component. All three models are related quantitativ…
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…
Since the end of 2019, COVID-19 has significantly affected the lives of people around the world. Towards the end of 2020, several COVID-19 vaccine candidates with relatively high efficacy have been reported in the final phase of clinical trials. Vaccines have been considered as critical tools for opening up social and economic activities, thereby lessening the impact of this disease on the society. This paper presents a simulation of COVID-19 spread using modified Susceptible-Infected-Removed (SIR) model under vaccine intervention in several localities of Malaysia, i.e. those cities or states …
Disease transmission is studied through disciplines like epidemiology, applied mathematics, and statistics. Mathematical simulation models for transmission have implications in solving public and personal health challenges. The SIR model uses a compartmental approach including dynamic and nonlinear behavior of transmission through three factors: susceptible, infected, and removed (recovered and deceased) individuals. Using the Lambert W Function, we propose a framework to study solutions of the SIR model. This demonstrates the applications of COVID-19 transmission data to model the spread of a…
We model and forecast the early evolution of the COVID-19 pandemic in Brazil using Brazilian recent data from February 25, 2020 to March 30, 2020. This early period accounts for unawareness of the epidemiological characteristics of the disease in a new territory, sub-notification of the real numbers of infected people and the timely introduction of social distancing policies to flatten the spread of the disease. We use two variations of the SIR model and we include a parameter that comprises the effects of social distancing measures. Short and long term forecasts show that the social distancin…
The global COVID-19 pandemic (SARS-CoV-2 virus) is the defining health crisis of our century. Due to the absence of vaccines and drugs that can help to fight it, the world solution to control the spread has been to consider public social distance measures that avoids the saturation of the health system. In this context, we investigate a Model Predictive Control (MPC) framework to determine the time and duration of social distancing policies. We use Brazilian data in the period from March to May of 2020. The available data regarding the number of infected individuals and deaths suffers from sub…
This paper develops an individual-based stochastic network SIR model for the empirical analysis of the Covid-19 pandemic. It derives moment conditions for the number of infected and active cases for single as well as multigroup epidemic models. These moment conditions are used to investigate the identification and estimation of the transmission rates. The paper then proposes a method that jointly estimates the transmission rate and the magnitude of under-reporting of infected cases. Empirical evidence on six European countries matches the simulated outcomes once the under-reporting of infected…
We study the relative importance of two key control measures for epidemic spreading: endogenous social self-distancing and exogenous imposed quarantine. We use the framework of adaptive networks, moment-closure, and ordinary differential equations (ODEs) to introduce several novel models based upon susceptible-infected-recovered (SIR) dynamics. First, we compare computationally expensive, adaptive network simulations with their corresponding computationally highly efficient ODE equivalents and find excellent agreement. Second, we discover that there exists a relatively simple critical curve in…
Social distancing and lockdown are the two main non-pharmaceutical interventions being used by the UK government to contain and control the COVID-19 epidemic; these are being applied uniformly across the entire country, even though the results of the Imperial College report by Ferguson et al show that the impact of the infection increases sharply with age. This paper develops a variant of the workhorse SIR model for epidemics, where the population is classified into a number of age groups. This allows us to understand the effects of age-dependent controls on the epidemic, and explore possible …
The presence of a large number of infected individuals with few or no symptoms is an important epidemiological difficulty and the main mathematical feature of COVID-19. The A-SIR model, i.e. a SIR (Susceptible-Infected-Removed) model with a compartment for infected individuals with no symptoms or few symptoms was proposed by Giuseppe Gaeta, arXiv:2003.08720 [q-bio.PE] (2020). In this paper we investigate a slightly generalized version of the same model and propose a scheme for fitting the parameters of the model to real data using the time series only of the deceased individuals. The scheme is…
In this paper, we conduct mathematical and numerical analyses to address the following crucial questions for COVID-19: (Q1) Is it possible to contain COVID-19? (Q2) When will be the peak and the end of the epidemic? (Q3) How do the asymptomatic infections affect the spread of disease? (Q4) What is the ratio of the population that needs to be infected to achieve herd immunity? (Q5) How effective are the social distancing approaches? (Q6) What is the ratio of the population infected in the long run? For (Q1) and (Q2), we propose a time-dependent susceptible-infected-recovered (SIR) model that tr…
The infectious diseases are spreading due to human interactions enabled by various social networks. Therefore, when a new pathogen such as SARS-CoV-2 causes an outbreak, the non-pharmaceutical isolation strategies (e.g., social distancing) are the only possible response to disrupt its spreading. To this end, we introduce the new epidemic model (SICARS) and compare the centralized (C), decentralized (D), and combined (C+D) social distancing strategies, and analyze their efficiency to control the dynamics of COVID-19 on heterogeneous complex networks. Our analysis shows that the centralized soci…
Network Behavioral-Feedback SIR Epidemic Model
该研究提出了一种网络行为反馈SIR流行病模型，分析了其平衡点稳定性与瞬态行为，拓展了感染曲线单峰性结果，为疫情控制策略设计提供了见解。
The rapid spread of the Coronavirus (COVID-19) confronts policy makers with the problem of measuring the effectiveness of containment strategies, balancing public health considerations with the economic costs of social distancing measures. We introduce a modified epidemic model that we name the controlled-SIR model, in which the disease reproduction rate evolves dynamically in response to political and societal reactions. An analytic solution is presented. The model reproduces official COVID-19 cases counts of a large number of regions and countries that surpassed the first peak of the outbrea…
Covid-19 is one of the most dreaded pandemics/epidemics in the world threatening the human population. The dynamics of this pandemic is quite complicated and prediction of pandemic states often fails. In this work, we study and correlate the SIR epidemiological model with the ongoing pandemic and found that pandemic dynamics and states are quite sensitively dependent on model parameters. The analysis of the exact parametric solution of the deterministic SIR model shows that the fixed points ($g^*$) depend on the SIR parameters, where, if $g^*\langle\fracγ{βx_0}$ then $g^*$ is stable and the pa…
We show how the standard field theoretical language based on creation and annihilation operators may be used for a straightforward derivation of an SIR-type stochastic model for COVID-19 epidemic, from which we obtain the time evolution of the mean number of infectious (active cases) and deceased individuals. In order to capture the effects of lockdown and social distancing, we use a time-dependent infection rate. The results are in good agreement with the data for three different waves of epidemic activity in South Korea.
In this research, we study the propagation patterns of epidemic diseases such as the COVID-19 coronavirus, from a mathematical modeling perspective. The study is based on an extensions of the well-known susceptible-infected-recovered (SIR) family of compartmental models. It is shown how social measures such as distancing, regional lockdowns, quarantine and global public health vigilance, influence the model parameters, which can eventually change the mortality rates and active contaminated cases over time, in the real world. As with all mathematical models, the predictive ability of the model …
As of December 2020, the COVID-19 pandemic has infected over 75 million people, making it the deadliest pandemic in modern history. This study develops a novel compartmental epidemiological model specific to the SARS-CoV-2 virus and analyzes the effect of common preventative measures such as testing, quarantine, social distancing, and vaccination. By accounting for the most prevalent interventions that have been enacted to minimize the spread of the virus, the model establishes a paramount foundation for future mathematical modeling of COVID-19 and other modern pandemics. Specifically, the mod…
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 integrat…
One of the main challenges of the measures against the COVID-19 epidemic is to reduce the amplitude of the epidemic peak without increasing without control its timescale. We investigate this problem using the SIR model for the epidemic dynamics, for which reduction of the epidemic peak $I_P$ can be achieved only at the price of increasing the time $t_P$ of its occurrence and its entire time-span $t_E$. By means of a time reparametrization we linearize the equations for the SIR dynamics. This allows us to solve exactly the dynamics in the time domain and to derive the scaling behaviour of the s…
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 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 …
Changes in behaviour when adherers to an intervention experience a different epidemic than non-adherers
本文研究了在流行病中，干预执行者与非执行者行为变化对疾病传播的影响，提出了一个结合行为和疾病动态的SIR框架模型，分析了不同策略下的感染情况。
The replacement number dynamics in SIR-type epidemic models I: From SSISS to RND picture
本文提出了SIR类流行病模型中的替换数动力学（RND）框架，通过SSISS纤维丛统一了多类流行病与社会行为模型，并证明其有界性与无周期解，为平衡点分析提供统一方法。
Motivated by the need for novel robust approaches to modelling the Covid-19 epidemic, this paper treats a population of $N$ individuals as an inhomogeneous random social network (IRSN). The nodes of the network represent different types of individuals and the edges represent significant social relationships. An epidemic is pictured as a contagion process that changes daily, triggered on day $0$ by a seed infection introduced into the population. Individuals' social behaviour and health status are assumed to be random, with probability distributions that vary with their type. First a formulatio…
The severe acute respiratory syndrome COVID-19 has been in the center of the ongoing global health crisis in 2020. The high prevalence of mild cases facilitates sub-notification outside hospital environments and the number of those who are or have been infected remains largely unknown, leading to poor estimates of the crude mortality rate of the disease. Here we use a simple model to describe the number of accumulated deaths caused by COVID-19. The close connection between the proposed model and an approximate solution of the SIR model provides a system of equations whose solutions are robust …
Standard epidemiological models for COVID-19 employ variants of compartment (SIR) models at local scales, implicitly assuming spatially uniform local mixing. Here, we examine the effect of employing more geographically detailed diffusion models based on known spatial features of interpersonal networks, most particularly the presence of a long-tailed but monotone decline in the probability of interaction with distance, on disease diffusion. Based on simulations of unrestricted COVID-19 diffusion in 19 U.S cities, we conclude that heterogeneity in population distribution can have large impacts o…
Here we propose and implement a generalized mathematical model to find the time evolution of population in infectious diseases and apply the model to study the recent COVID-19 pandemic. Our model at the core is a non-local generalization of the widely used Kermack-McKendrick(KM) model where the susceptible(S) population evolves into two other categories, namely infectives(I) and removed(R). This is the well-known SIR model in which we further divide both S and I into high and low risk categories. We first formulate a set of non-local dynamical equations for the time evolution of distinct popul…
The COVID-19 pandemic is the profoundest health crisis of the 21rst century. The SARS-CoV-2 virus arrived in Brazil around March, 2020 and its social and economical backlashes are catastrophic. In this paper, it is investigated how Model Predictive Control (MPC) could be used to plan appropriate social distancing policies to mitigate the pandemic effects in Bahia and Santa Catarina, two states of different regions, culture, and population demography in Brazil. In addition, the parameters of Susceptible-Infected-Recovered-Deceased (SIRD) models for these two states are identified using an optim…
The last decade saw the advent of increasingly realistic epidemic models that leverage on the availability of highly detailed census and human mobility data. Data-driven models aim at a granularity down to the level of households or single individuals. However, relatively little systematic work has been done to provide coupled behavior-disease models able to close the feedback loop between behavioral changes triggered in the population by an individual's perception of the disease spread and the actual disease spread itself. While models lacking this coupling can be extremely successful in mild…
This study develops a framework for quantification of the impact of changes in population mobility due to social distancing on the COVID-19 infection growth rate. Using the Susceptible-Infected-Recovered (SIR) epidemiological model we establish that under some mild assumptions the growth rate of COVID-19 deaths is a time-delayed approximation of the growth rate of COVID-19 infections. We then hypothesize that the growth rate of COVID-19 infections is a function of population mobility, which leads to a statistical model that predicts the growth rate of COVID-19 deaths as a delayed function of p…
Opinion-driven risk perception and reaction in SIS epidemics
本文构建观点驱动接触率的SIS耦合模型，证明高低基线传染性两类参数区间及双稳态，并揭示风险寻求、风险规避和高反应性对感染水平及根除的影响。
During pandemic events, strategies such as social distancing can be fundamental to curb viral spreading. Such actions can reduce the number of simultaneous infections and mitigate the disease spreading, which is relevant to the risk of a healthcare system collapse. Although these strategies can be suggested, their actual implementation may depend on the population perception of the disease risk. The current COVID-19 crisis, for instance, is showing that some individuals are much more prone than others to remain isolated, avoiding unnecessary contacts. With this motivation, we propose an epidem…
The detection and management of diseases become quite complicated when pathogens contain asymptomatic phenotypes amongst their ranks, as evident during the recent COVID-19 pandemic. Spreading of diseases has been studied extensively under the paradigm of Susceptible - Infected - Recovered - Deceased (SIRD) dynamics. Various game-theoretic approaches have also addressed disease spread, many of which consider S, I, R, and D as strategies rather than as states. Remarkably, most studies from the above approaches do not account for the distinction between the symptomatic or asymptomatic aspect of t…
We are currently living in a state of uncertainty due to the pandemic caused by the Sars-CoV-2 virus. There are several factors involved in the epidemic spreading such as the individual characteristics of each city/country. The true shape of the epidemic dynamics is a large, complex system such as most of the social systems. In this context, Complex networks are a great candidate to analyze these systems due to their ability to tackle structural and dynamical properties. Therefore this study presents a new approach to model the COVID-19 epidemic using a multi-layer complex network, where nodes…
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