SIR Epidemic Model Research Articles

Unraveling the role of adapting risk perception during the COVID-19 pandemic in Europe

During the COVID-19 pandemic, the behavioral response to reported case numbers changed drastically over time. While a few dozen cases were enough to trigger government-induced and voluntary contact reduction in early 2020, less than a year later much higher case numbers were required to induce behavioral change. Little attention has been paid to understand and mathematically model this effect of decreasing risk perception over longer time-scales. Here, first we show that weighing the number of cases with a time-varying factor of the form ta,a<0 explains real-world mobility patterns from several European countries during 2020 when introduced into a very simple behavior model. Subsequently, we couple our behavior model with an SIR epidemic model. Remarkably, decreasing risk perception can produce complex dynamics, including multiple waves of infection. We find two regimes for the total number of infected individuals that are explained by the interplay of initial attention and the rate of attention decrease. Our results show that including adaption into non-equilibrium models is necessary to understand behavior change over long time scales and the emergence of non-trivial infection dynamics.

Fractional-order SIR epidemic model with treatment cure rate

In this paper, we will study a fractional-order SIR epidemic model with treatment cure rate. The paper starts with the investigation of some needed preliminaries related to fractional calculus. Next, we investigate the well-posedness of the fractional-order model in terms of positivity and boundedness of solution. Then, we give the basic reproduction number via the next generation matrix method. The model has two steady states, namely, the disease-free equilibrium and the endemic equilibrium. It was proven that when the basic reproduction number is less than unity, then, the disease-free-equilibrium is globally asymptotically stable. While, when the basic reproduction number is greater than unity, then the endemic-equilibrium is globally asymptotically stable. Numerical simulations are carried out in order to confirm the theoretical results. To this end, we will apply a numerical approach based on the fundamental theorem of fractional calculus and a three-step Lagrange polynomial interpolation technique. It is shown that any increase of the fractional order derivative leads to a decrease in the convergence speed towards the studied equilibrium.

Matrix Product Belief Propagation for reweighted stochastic dynamics over graphs.

Stochastic processes on graphs can describe a great variety of phenomena ranging from neural activity to epidemic spreading. While many existing methods can accurately describe typical realizations of such processes, computing properties of extremely rare events is a hard task, particularly so in the case of recurrent models, in which variables may return to a previously visited state. Here, we build on the matrix product cavity method, extending it fundamentally in two directions: First, we show how it can be applied to Markov processes biased by arbitrary reweighting factors that concentrate most of the probability mass on rare events. Second, we introduce an efficient scheme to reduce the computational cost of a single node update from exponential to polynomial in the node degree. Two applications are considered: inference of infection probabilities from sparse observations within the SIRS epidemic model and the computation of both typical observables and large deviations of several kinetic Ising models.

Ranking influential nodes in complex network using edge weight degree based shell decomposition

Identifying critical nodes in a complex network or ranking the nodes based on their influence over the network has many utilities. As an example, these come in handy while choosing customers for viral marketing of a new product or identifying users whom to block for preventing the spreading of misinformation/rumor among others. With proper network modeling, a large number of real-world problems could be analyzed using such node ranking methods. Ranking the nodes based on their spreading influence in a complex network is an important research problem. Network topology-based techniques play a significant role in various applications that require knowing the central entities in a modeled system. Classical network centralities like degree, betweenness, and closeness followed by core decomposition techniques are applied successfully in ranking nodes. Each of them having certain limitations like purely local measure or computation heavy etc. Due to the inherent nature of the real-world problems/events, when they are modeled as a network, most of them transform into a weighted network. Even then a large number of literature is focused on unweighted networks due to the simplicity of computations and evaluation metrics. Many node ranking methods developed for unweighted networks may be employed to weighted networks but they will not be utilize the extra information available as connection strengths. Recently, many authors have started to extend the node ranking heuristics for the weighted networks. In this paper, we propose an edge weight degree neighborhood (EwDN) based shell decomposition technique to rank the nodes of a weighted network. It considers both node degree and edge weight with a proper balance of importance. The SIR epidemic model has been used as a benchmark for simulating the spreading process. The number of generated shells, number of core–shell nodes, and monotonicity have been computed as performance metrics during the evaluation of the methods. Also, Kendall’s rank correlation with the benchmark SIR model for various competing heuristics is compared. A performance comparison with recent relevant methods shows a better outcome with the proposed method on real-world weighted networks.

Threshold dynamics of an uncertain SIRS epidemic model with a bilinear incidence

This paper mainly discusses the extinction and persistent dynamic behavior of infectious diseases with temporary immunity. Considering that the transmission process of infectious diseases is affected by environmental fluctuations, stochastic SIRS models have been proposed, while the outbreak of diseases is sudden and the interference terms that affect disease transmission cannot be qualified as random variables. Liu process is introduced based on uncertainty theory, which is a new branch of mathematics for describing uncertainty phenomena, to describe uncertain disturbances in epidemic transmission. This paper first extends the classic SIRS model from a deterministic framework to an uncertain framework and constructs an uncertain SIRS infectious disease model with constant input and bilinear incidence. Then, by means of Yao-Chen formula, α-path of uncertain SIRS model and the corresponding ordinary differential equations are obtained to introduce the uncertainty threshold function R 0 * as the basic reproduction number. Moreover, two equilibrium states are derived. A series of numerical examples show that the larger the value of R 0 * , the more difficult it is to control the disease. If R 0 * ≤ 1 , the infectious disease will gradually disappear, while if R 0 * &gt; 1 , the infectious disease will develop into a local epidemic.

Global Asymptotic Stability of a Hybrid Differential–Difference System Describing SIR and SIS Epidemic Models with a Protection Phase and a Nonlinear Force of Infection

Global Asymptotic Stability of a Hybrid Differential–Difference System Describing SIR and SIS Epidemic Models with a Protection Phase and a Nonlinear Force of Infection

Dynamics of Competitive Two-Strain Stochastic SIR Epidemics on Heterogeneous Networks

Mathematical modeling in epidemiology, biology, and life sciences requires the use of stochastic models. In this paper, we derive a competitive two-strain stochastic SIR epidemic model by considering the change in state of the epidemic process due to an event. Based on the density-dependent process theory, we construct a six-dimensional deterministic model that can be used to describe the diffusion limit of the stochastic epidemic on a heterogeneous network. Furthermore, we show the explicit expressions for the variances of infectious individuals with strain 1 and strain 2 when the level of infection is increasing exponentially. In particular, we find that the expressions of the variances are symmetric. Finally, simulations for epidemics spreading on networks are performed to confirm our analytical results. We find a close agreement between the simulations and theoretical predictions.

Complex Dynamics of a Stochastic SIR Epidemic Model with Vertical Transmission and Varying Total Population Size

Complex Dynamics of a Stochastic SIR Epidemic Model with Vertical Transmission and Varying Total Population Size

Dynamic analysis of a fractional order SIR model with specific functional response and Holling type II treatment rate

This paper investigates a fractional order SIR epidemic model with specific functional response and Holling type II treatment rate with fractional derivative in the Caputo sense. A thorough investigation of the existence, uniqueness, non-negativity and boundedness of the solutions has been performed in the beginning. Stability of the equilibria is analysed in the section next to it. Bifurcation analysis has also been carried out for the model. Finally, numerical simulations are used to support the theoretical findings.

Research on risk contagion mechanism of big fintech based on the SIRS model.

In recent years, while the relationship between the new financial institutions, represented by financial technology companies, and the traditional financial institutions(banks, securities, insurance, etc.) has been steadily enhanced, a New Relational Network has silently emerged. Along with the rapid expansion of big fintech companies, the possibility of financial risk breeding and spreading in the New Relational Network is also rising. This article analyzes and simulates the risk contagion mechanism of big fintech risks based on the SIRS epidemic model. The study's findings imply that: when the number of big fintech companies infected with risk exceeds the risk threshold, the big fintech risk will spread in the New Relational Network. At this time, the number of big fintech companies infected with risk can be reduced below the threshold by enhancing the risk warning, risk management, risk buffering and blocking capabilities, and timely improving risk prevention and control measures in the post-infection phase. It means that the big fintech risk is controlled. For big fintech risks, proactive interventions are more effective than post-incident response measures. This paper makes the following recommendations for preventing big fintech risks: creating a risk monitoring and early warning system to raise the Big Fintech companies' direct immunization rates; strengthening the big fintech companies' risk management and risk mitigation capabilities; enhancing the internal and external supervision to achieve sustainable development of big fintech companies.

Collective epidemics with asymptomatics and functional infection rates

ABSTRACT This paper discusses a generalized SIR epidemic model that incorporates infectives with or without symptoms, allows arbitrary distributions for infectious periods and assumes infection rates depending on the current size of susceptibles. Our interest lies in the joint distribution of the state of the population and the severity of the disease at the end of infection. The approach is based on a formulation of the epidemic as a so-called collective model. First, a set of martingales is constructed which provides the distribution of this final epidemic outcome. Then, its corresponding distributional transform is expressed in terms of a family of pseudo-polynomials of Abel-Gontcharoff type. Some of the results obtained are illustrated numerically.

Biochemical models of SIR and SIRS: Effects of bilinear incidence rate on infection-free and endemic states.

Understanding and forecasting the progression of disease epidemics is possible through the study of nonlinear epidemic biochemical models that describe the relationship among susceptible, infected, and immune individuals in a population. In this paper, by determining the algebraic invariant planes and studying the Hopf bifurcation on these invariant planes, we study the stability of the Hopf bifurcation in the infection-free and endemic states of the SIR and SIRS epidemic models with bilinear incidence rate. We analyze the stability of the limit cycles of the bilinear incidence SIR and SIRS models at the steady state point where infection vanishes and at the endemic steady state point where the system behaves in an oscillatory manner. We demonstrate the algebraic results by numerical simulations for parameter values that satisfy the conditions for both free and endemic states.

Qualitative analysis of a reaction-diffusion SIRS epidemic model with nonlinear incidence rate and partial immunity.

In this paper, a reaction-diffusion SIRS epidemic model with nonlinear incidence rate and partial immunity in a spatially heterogeneous environment is proposed. The well-posedness of the solution is firstly established. Then the basic reproduction number R0 is defined and a threshold dynamics is obtained. That is, when R0<1, the disease-free steady state is locally stable, which implies that the disease is extinct, when R0>1, the disease is permanent, and there exists at least one positive steady state solution. Finally, the asymptotic profiles of the positive steady state solution as individuals disperse at small and large rates are investigated. Furthermore, as an application of theoretical analysis, a numerical example involving the spread of influenza is discussed. Based on the numerical simulations, we find that the increase of transmission rate and spatial heterogeneity can enhance the risk of influenza propagation, and the increase of diffusion rate, saturation incidence for susceptible and recovery rate can reduce the risk of influenza propagation. Therefore, we propose to reduce the flow of people to lower the effect of spatial heterogeneity, increase the transfer of infected individuals to hospitals in surrounding areas to increase the diffusion rate, and increase the construction of public medical resources to increase the recovery rate for controlling influenza propagation.

Effect of disease-induced death rate and latent period on global stability for SIRS epidemic models with general incidence rate

Effect of disease-induced death rate and latent period on global stability for SIRS epidemic models with general incidence rate

Optimal control of a spatiotemporal [formula omitted] model with reaction–diffusion involving [formula omitted]-Laplacian operator

The present paper’s primary goal is to make an analysis and study a reaction–diffusion SIR epidemic mathematical model expressed as a parabolic system of partial differential equations using the p-Laplacian operator. Immunity is compelled through vaccination distribution, which is seen as a control variable. Our main goal is to define an optimal control, which reduces the spread of infection and the cost of vaccination over a limited period of time and space. Existence and uniqueness of a positive solution and existence of an optimal control for the proposed model are proved. Then a description and characterization of the optimal control is provided in terms of state and adjoint functions. Optimality system is numerically resolved by a discrete iterative scheme pertained to and the forward–backward algorithm. Furthermore, using various p-values for the p-Laplacian operator, numerical results demonstrate the effectiveness of the suggested control strategy, which yields meaningful outcomes.

Epidemic spreading dynamics on two-layer complex networks

Recent studies have shown that many real-world systems can be described by multi-layer complex networks. In this paper, the concept of layers is introduced to construct a traffic-driven SIR epidemic spreading model on “logical-physical” layered network. Based on the peak density of infected nodes and the ultimate density of recovered nodes, we investigate the features of epidemic spreading on layered network. Through numerical simulations, it is shown that traffic flow greatly influences the intensity and scope of epidemic spreading. By comparing the effects of four kinds of two-layer networks generated by ER random network model and BA scale-free network model on epidemic spreading, we found that the homogeneity of logical or physical network structure can promote the spread of epidemic more than heterogeneous networks. This work may be of service to design traffic-driven epidemic prevention and control strategies.

Dynamics of a nonlocal SIR epidemic model with free boundaries

This paper is devoted to a nonlocal SIR type epidemic model, with free boundary denoting the expanding front. We aim to examine the full information about the dynamical behaviors of the involved disease that caused by the nonlocal diffusion and growing habitat. We obtain a classification for the spreading and vanishing cases, and show that those are determined by the new proposed threshold function R0F(t) and especially the corresponding initial data R0F(0). We prove that R0F(t)≤R0 and R0F(t)→R0 as t tends to ∞, R0 here representing the basic reproductive number of the associated ordinary differential equation. Also, the result show that the introduce of the nonlocal diffusion makes the disease becomes more easier to spread. In addition, we present some numerical simulations to confirm the theoretical results, as well as the impacts of R0 and the expanding capability.

An Efficient Approach to Solving the Fractional SIR Epidemic Model with the Atangana–Baleanu–Caputo Fractional Operator

In this article, we study the fractional SIR epidemic model with the Atangana–Baleanu–Caputo fractional operator. We explore the properties and applicability of the ZZ transformation on the Atangana–Baleanu–Caputo fractional operator as the ZZ transform of the Atangana–Baleanu–Caputo fractional derivative. This study is an application of two power methods. We obtain a special solution with the homotopy perturbation method (HPM) combined with the ZZ transformation scheme; then we present the problem and study the existence of the solution, and also we apply this new method to solving the fractional SIR epidemic with the ABC operator. The solutions show up as infinite series. The behavior of the numerical solutions of this model, represented by series of the evolution in the time fractional epidemic, is compared with the Adomian decomposition method and the Laplace–Adomian decomposition method. The results showed an increase in the number of immunized persons compared to the results obtained via those two methods.

Asymptotic behavior of an epidemic model with infinitely many variants.

We investigate the long-time dynamics of a SIR epidemic model with infinitely many pathogen variants infecting a homogeneous host population. We show that the basic reproduction number [Formula: see text] of the pathogen can be defined in that case and corresponds to a threshold between the persistence ([Formula: see text]) and the extinction ([Formula: see text]) of the pathogen population. When [Formula: see text] and the maximal fitness is attained by at least one variant, we show that the systems reaches an endemic equilibrium state that can be explicitly determined from the initial data. When [Formula: see text] but none of the variants attain the maximal fitness, the situation is more intricate. We show that, in general, the pathogen is uniformly persistent and any family of variants that have a fitness which is uniformly lower than the optimal fitness, eventually gets extinct. We derive a condition under which the total pathogen population converges to a limit which can be computed explicitly. We also find counterexamples that show that, when our condition is not met, the total pathogen population may converge to an unexpected value, or the system can even reach an eternally transient behavior where the total pathogen population between several values. We illustrate our results with numerical simulations that emphasize the wide variety of possible dynamics.

A highly accurate peak time formula of epidemic outbreak from the SIR model

Forecasting the epidemic peak time right from the origination of a disease is vital to take over dynamical behaviour of its spread over time. The decision of isolation, social distance and lock down strategic progresses does all rely on an accurate prediction of the peak time so that reduction of the time of peak or of the infected size of population will be made possible. Therefore, recent efforts concentrated on deriving elaborative and analytically accessible expressions representing the peak time of the infected compartment from the classical SIR epidemic mathematical model. In this research paper, two closed-form formulae are introduced to yield a straightforward computation of peak time of an infectious disease with no restrictions on the SIR quantities. In addition to this, the calculations can be implemented on a usual calculator, without requiring the use of advanced mathematical functions, having provided the initial fractions of infected and susceptible populations as well as the recovery to infectious ratio. A comparison including the COVID-19 data is fulfilled with the very recent formulas available in the open literature. With the proposed new scalings, evaluation of the peak time is reduced only to two parameter space and the accuracy of the present formulas in reduced form is ultimately confirmed yielding an error of order of magnitude 10−4 valid for the complete regime of the set of SIR model parameters. Even in the case of an endemic, the past peak time of the illness can also be captured accurately by the given formulae. Two simple approximations in terms of usual geometric series are also provided. These can be safely used with a pocket calculator without sophisticated laboratory equipments.