Susceptible-infected-recovered Epidemic Model Research Articles

Solution of the SIR epidemic model of arbitrary orders containing Caputo-Fabrizio, Atangana-Baleanu and Caputo derivatives

<abstract><p>The main aim of this study was to apply an analytical method to solve a nonlinear system of fractional differential equations (FDEs). This method is the Adomian decomposition method (ADM), and a comparison between its results was made by using a numerical method: Runge-Kutta 4 (RK4). It is proven that there is a unique solution to the system. The convergence of the series solution is given, and the error estimate is also proven. After that, the susceptible-infected-recovered (SIR) model was taken as an real phenomenon with such systems. This system is discussed with three different fractional derivatives (FDs): the Caputo-Fabrizio derivative (CFD), the Atangana-Baleanu derivative (ABD), and the Caputo derivative (CD). A comparison between these three different derivatives is given. We aimed to see which one of the new definitions (ABD and CFD) is close to one of the most important classical definitions (CD).</p></abstract>

Traffic-driven SIR epidemic spread dynamics on scale-free networks

Traffic flow affects the transmission and distribution of pathogens. The large-scale traffic flow that emerges with the rapid development of global economic integration plays a significant role in the epidemic spread. In order to more accurately indicate the time characteristics of the traffic-driven epidemic spread, new parameters are added to represent the change of the infection rate parameter over time on the traffic-driven Susceptible–Infected–Recovered (SIR) epidemic spread model. Based on the collected epidemic data in Hebei Province, a linear regression method is performed to estimate the infection rate parameter and an improved traffic-driven SIR epidemic spread dynamics model is established. The impact of different link-closure rules, traffic flow and average degree on the epidemic spread is studied. The maximum instantaneous number of infected nodes and the maximum number of ever infected nodes are obtained through simulation. Compared to the simulation results of the links being closed between large-degree nodes, closing the links between small-degree nodes can effectively inhibit the epidemic spread. In addition, reducing traffic flow and increasing the average degree of the network can also slow the epidemic outbreak. The study provides the practical scientific basis for epidemic prevention departments to conduct traffic control during epidemic outbreaks.

On [formula omitted]-manifolds stability for impulsive delayed SIR epidemic models

In this paper, we study an impulsively extended delayed SIR (Susceptible-Infected-Recovered) epidemic model for the spread of infectious diseases. The impulsive control model is represented by a neural network system with reaction-diffusion terms and impulses at fixed instants of time. The notion of stability of specific manifolds defined by continuous functions is introduced to the model under consideration. Using the Lyapunov impulsive approach, we derive criteria for the global practical exponential stability of the defined manifolds of solutions. Since the stability of manifolds concepts generalize the stability of separate state notions, our results are more general and they extend some existing stability results for non-impulsive and impulsive SIR epidemic models. An example is considered to show the effectiveness of our results.

Positivity-Preserving Numerical Method for a Stochastic Multi-Group SIR Epidemic Model

Abstract The stochastic multi-group susceptible–infected–recovered (SIR) epidemic model is nonlinear, and so analytical solutions are generally difficult to obtain. Hence, it is often necessary to find numerical solutions, but most existing numerical methods fail to preserve the nonnegativity or positivity of solutions. Therefore, an appropriate numerical method for studying the dynamic behavior of epidemic diseases through SIR models is urgently required. In this paper, based on the Euler–Maruyama scheme and a logarithmic transformation, we propose a novel explicit positivity-preserving numerical scheme for a stochastic multi-group SIR epidemic model whose coefficients violate the global monotonicity condition. This scheme not only results in numerical solutions that preserve the domain of the stochastic multi-group SIR epidemic model, but also achieves the “ order - 1 2 {\mathrm{order}-\frac{1}{2}} ” strong convergence rate. Taking a two-group SIR epidemic model as an example, some numerical simulations are performed to illustrate the performance of the proposed scheme.

The Stochastic Approach for SIR Epidemic Models: Do They Help to Increase Information from Raw Data?

The recent outbreak of COVID-19 underlined the need for a fast and trustworthy methodology to identify the features of a pandemic, whose early identification is of help for designing non-pharmaceutical interventions (including lockdown and social distancing) to limit the progression of the disease. A common approach in this context is the parameter identification from deterministic epidemic models, which, unfortunately, cannot take into account the inherent randomness of the epidemic phenomenon, especially in the initial stage; on the other hand, the use of raw data within the framework of a stochastic model is not straightforward. This note investigates the stochastic approach applied to a basic SIR (Susceptible, Infected, Recovered) epidemic model to enhance information from raw data generated in silico. The stochastic model consists of a Continuous-Time Markov Model, describing the epidemic outbreak in terms of stochastic discrete infection and recovery events in a given region, and where independent random paths are associated to different provinces of the same region, which are assumed to share the same set of model parameters. The estimation procedure is based on the building of a loss function that symmetrically weighs first-order and second-order moments, differently from the standard approach that considers a highly asymmetrical choice, exploiting only first-order moments. Instead, we opt for an innovative symmetrical identification approach which exploits both moments. The new approach is specifically proposed to enhance the statistical information content of the raw epidemiological data.

Accurate Numerical Treatment on a Stochastic SIR Epidemic Model with Optimal Control Strategy

In this paper, a numerical study has been undertaken on the susceptible-infected-recovered (SIR) epidemic model that encompasses the mechanisms of the evolution of disease transmission; a prophylactic vaccination strategy in the susceptible populations, depending on the infective individuals. We furnish numerical and graphical simulation combined with explicit series solutions of the proposed model using the New Iterative Method (NIM) and Modified New Iterative Method (MNIM). The analytic-numeric New Iterative Method failed to deliver accurate solution for the large time domain. A new reliable algorithm based on NIM, the coupling of the Laplace transforms, and the New Iterative method is called Modified New Iterative Method (MNIM) which is presented to enhance the validity domain of NIM techniques. The convergence analysis of the MNIM has also been illustrated. The simulation results show that the vaccination strategy can slow down the spread of the epidemic rapidly. Numerical results illustrate the excellent performance of the MNIM and show that the modified method is much more accurate than the NIM.

Concept and solution of digital twin based on a Stieltjes differential equation

We introduce the concept of a digital twin by using Stieltjes differential equations. A precise mathematical definition of solution to the problem is presented. We also analyze the existence and uniqueness of solutions, introduce the concept of main digital twin. As a particular case, the classical compartmental SIR (susceptible, infected, recovered) epidemic model is considered and we study the interrelation between the digital twin and the system. In doing so, we use Stieltjes derivatives to feed the data from the real system to the virtual model which, in return, improves it in real time. Numerical simulations with real data of the COVID‐19 epidemic show the accuracy of the proposed ideas.

Fractional-Order Discrete-Time SIR Epidemic Model with Vaccination: Chaos and Complexity

This research presents a new fractional-order discrete-time susceptible-infected-recovered (SIR) epidemic model with vaccination. The dynamical behavior of the suggested model is examined analytically and numerically. Through using phase attractors, bifurcation diagrams, maximum Lyapunov exponent and the 0−1 test, it is verified that the newly introduced fractional discrete SIR epidemic model vaccination with both commensurate and incommensurate fractional orders has chaotic behavior. The discrete fractional model gives more complex dynamics for incommensurate fractional orders compared to commensurate fractional orders. The reasonable range of commensurate fractional orders is between γ = 0.8712 and γ = 1, while the reasonable range of incommensurate fractional orders is between γ2 = 0.77 and γ2 = 1. Furthermore, the complexity analysis is performed using approximate entropy (ApEn) and C0 complexity to confirm the existence of chaos. Finally, simulations were carried out on MATLAB to verify the efficacy of the given findings.

Impact of media coverage on a fractional-order SIR epidemic model

In this work, we formulated and analyzed a fractional-order epidemic model of infectious disease (such as SARS, 2019-nCoV and COVID-19) concerning media effect. The model is based on classical susceptible-infected-recovered (SIR) model. Basic properties regarding positivity, boundedness and non-negative solutions are discussed. Basic reproduction number [Formula: see text] of the system has been calculated using next-generation matrix method and it is seen that the disease-free equilibrium is locally as well as globally asymptotically stable if [Formula: see text], otherwise unstable. The existence of endemic equilibrium point is established using the Lambert W function. The condition for global stability has been derived. Numerical simulation suggests that fractional order and media have a large effect on our system dynamics. When media impact is stronger enough, our fractional-order system stabilizes the oscillation.

An algorithm for the direct estimation of the parameters of the SIR epidemic model from the I(t) dynamics.

The discrete SIR (Susceptible–Infected–Recovered) model is used in many studies to model the evolution of epidemics. Here, we consider one of its dynamics—the exponential decrease in infected cases I(t). By considering only the I(t) dynamics, we extract three parameters: the exponent of the initial exponential increase gamma ; the maximum value I_{max }; and the exponent of the final decrease varGamma . From these three parameters, we show mathematically how to extract all relevant parameters of the SIR model. We test this procedure on numerical data and then apply the methodology to real data received from the COVID-19 situation in France. We conclude that, based on the hospitalized data and the ICU (Intensive Care Unit) cases, two exponentials are found, for the initial increase and the decrease in I(t). The parameters found are larger than reported in the literature, and they are associated with a susceptible population which is limited to a sub-sample of the total population. This may be due to the fact that the SIR model cannot be applied to the covid-19 case, due to its strong hypotheses such as mixing of all the population, or also to the fact that the parameters have changed over time, due to the political initiatives such as social distanciation and lockdown.

Controlling Infectious Diseases: The Decisive Phase Effect on a Seasonal Vaccination Strategy

The study of epidemiological systems has generated deep interest in exploring the dynamical complexity of common infectious diseases driven by seasonally varying contact rates. Mathematical modeling and field observations have shown that, under seasonal variation, the incidence rates of some endemic infectious diseases fluctuate dramatically and the dynamics is often characterized by chaotic oscillations in the absence of specific vaccination programs. In fact, the existence of chaotic behavior has been precisely stated in the literature as a noticeable feature in the dynamics of the classical Susceptible-Infected-Recovered (SIR) seasonally forced epidemic model. However, in the context of epidemiology, chaos is often regarded as an undesirable phenomenon associated with the unpredictability of infectious diseases. As a consequence, the problem of converting chaotic motions into regular motions becomes particularly relevant. In this article, we consider the so-called phase control method applied to the seasonally forced SIR epidemic model to suppress chaos. Interestingly, this method of controlling chaos has a clear meaning as a weak perturbation on a seasonal vaccination strategy. Numerical simulations show that the phase difference between the two periodic forces — contact rate and vaccination — plays a very important role in controlling chaos.

Robust static output-feedback vaccination policy design for an uncertain SIR epidemic model with disturbances: Positive Takagi–Sugeno model approach

An analytic vaccination policy is established for a nonlinear susceptible–infected–recovered (SIR) epidemic model involving uncertainties and disturbances by using only the inaccurate information of the infected through the positive Takagi–Sugeno (T–S) fuzzy approach. The SIR model is exactly represented as the fuzzy model in the T–S form. Two conditions for fuzzy static output-feedback controllers are formulated using the symmetric and diagonal Lyapunov matrices. These controllers exhibit L∞–L∞ disturbance attenuation performance, are robust against norm-bounded parametric uncertainties, and preserve closed-loop positivity. The case study for the positive stabilization of the uncertain nonlinear SIR model demonstrates the effectiveness of the proposed methodology.

Qualitative property preservation of high-order operator splitting for the SIR model

The susceptible-infected-recovered (SIR) model is perhaps the most basic epidemiological model for the evolution of disease spread within a population. Because of its direct representation of fundamental physical quantities, a true solution to an SIR model possesses a number of qualitative properties, such as conservation of the total population or positivity or monotonicity of its constituent populations, that may only be guaranteed to hold numerically under step-size restrictions on the solver. Operator-splitting methods with order greater than two require backward sub-steps in each operator, and the effects of these backward sub-steps on the step-size restrictions for guarantees of qualitative correctness of numerical solutions are not well studied. In this study, we analyze the impact of backward steps on step-size restrictions for guaranteed qualitative properties by applying third- and fourth-order operator-splitting methods to the SIR epidemic model. We find that it is possible to provide step-size restrictions that guarantee qualitative property preservation of the numerical solution despite the negative sub-steps, but care must be taken in the choice of the method. Results such as this open the door for the design and application of high-order operator-splitting methods to other mathematical models in general for which qualitative property preservation is important.

Quasi-stationary probability distribution for the SIR epidemic model

In this paper, we propose two new approximations to the joint quasi-stationary distribution (QSD) of the number susceptible and infected individuals in the Susceptible-Infected-Recovered (SIR) stochastic epidemic model and we derive the marginal QSD of the infected individuals. These two approximations depend on the basic reproduction number [Formula: see text] and give a positive probability of the QSD to all the transient states. Numerical comparisons are presented to check the accuracy of these approximations.

A chaos study of fractional SIR epidemic model of childhood diseases

Models of bio-mathematics are experimental systems that recreate aspects of human tissue function, diseases or virus. In this research, a new operational matrix based on the Laguerre wavelets is introduced for a arbitrary-order susceptible-infected-recovered (SIR) epidemic dynamical system of childhood diseases. An exact mechanism for the Riemann–Liouville arbitrary integral operator for the Laguerre wavelets is explained where the arbitrary-order derivative is assumed in the Liouville-Caputo style. Further, we use this operational matrix to convert the given dispute into a system of algebraic equations. The chaotic attractors for fractional-order SIR dynamical model are illustrated graphically by adopting the Adams–Bashforth-Moulton (ABM) scheme. Numerical simulations and results for the susceptible, infected and recovered peoples are carried out by using the Laguerre wavelets. Their behaviour with respect to time is seen to be the key features of this work. Moreover, we have compared the Laguerre wavelet solutions with the ABM solution for the truthfulness and applicability of the Laguerre wavelets scheme.

Numerical Solution of an Interval-Based Uncertain SIR (Susceptible–Infected–Recovered) Epidemic Model by Homotopy Analysis Method

This work proposes an interval-based uncertain Susceptible–Infected–Recovered (SIR) epidemic model. The interval model has been numerically solved by the homotopy analysis method (HAM). The SIR epidemic model is proposed and solved under different uncertain intervals by the HAM to obtain the numerical solution of the model. Furthermore, the SIR ODE model was transformed into a stochastic differential equation (SDE) model and the results of the stochastic and deterministic models were compared using numerical simulations. The results obtained were compared with the numerical solution and found to be in good agreement. Finally, various simulations were done to discuss the solution.

Stability Analysis of Fractional SIR Model Related to Delay in State and Control Variables

The study of a nonlinear mathematical fractional SIR (Susceptible - Infected - Recovered) epidemiological model related to the delay in state and control variables in terms of time is the focus of this paper. The existence of a bounded solution for the fractional SIR epidemic model has been demonstrated, and it is unique. A new set of infection-free equilibrium points has been discovered, and their local stability has been investigated. In addition, using the next-generation matrix method, the basic reproductive number Ro was calculated

Model Penyebaran Penyakit SIR Tipe Rantai Binomial dengan Kontak Random dan Waktu Penyembuhan Bernilai Tak Hingga

<p>Susceptible-Infected-Recovered (SIR) epidemic model is an epidemic model that illustrates the pattern of disease spread with the characteristics of individuals who have recovered cannot be re-infected and have a permanent immune system. The binomial chain type epidemic model assumes that infection spreads in discrete time units and the number of the infected individuals follows a binomial distribution. This research aims to discuss binomial chain type SIR epidemic model by simulating the model. The transition probability depends on the number of infected individuals in the period the number of individuals encountered, and the transmission probability. This model also assumes an infinite recovery time ( = ∞). This situation illustrates that infected individuals remain contagious during the period of spread of the disease. This situation can arise when the causative agent of the disease has a long life. Then simulations are performed by giving different transmission probability The results show that the greater transmission probability will cause the probability of a new individual being infected in the next period to be greater.</p><p><strong>Keywords</strong><strong> : </strong>SIR<em> </em>epidemic model, binomial chain, infinite recovery time</p>

Predicting Oil Production Sites for Planning Road Infrastructure: Trip Generation Using SIR Epidemic Model

Drilling activity produces a significant amount of road traffic through unpaved and paved local roads. Because oil production is an important contributor to the local economy in the state of North Dakota, the state and local transportation agencies make efforts to support local energy logistics through the expansion and good repair and maintenance of transportation infrastructure. As part of this effort, it is important to build new roads and bridges, maintain existing road pavement and non-marked road surface conditions, and improve bridge and other transportation infrastructure. Therefore, the purpose of this study is to review previous oil location prediction models and propose a novel geospatial model to predict drilling locations which have a significant impact on local roads, to verify and provide a better prediction model. Then, this study proposes a SIR (susceptible–infected–recovered) epidemic model to predict oil drilling locations which are traffic generators. The simulation has been done on the historical data from 1980 to 2015. The study found that the best fit parameters of β (contact rate) and μ (recovery rate) were estimated by using a dataset of historical oil wells. The study found that the SIR epidemic model can be applied to predict the locations of oil wells. The proposed model can be used to predict other drilling locations and can assist with traffic, road conditions, and other related issues, which is a much needed predictive model that is key in transportation planning and pavement design and maintenance.

Evaluation of Epidemic-Based Information Dissemination in a Wireless Network Testbed

Information dissemination is an integral part of modern networking environments, such as Wireless Sensor Networks (WSNs). Probabilistic flooding, a common epidemic-based approach, is used as an efficient alternative to traditional blind flooding as it minimizes redundant transmissions and energy consumption. It shares some similarities with the Susceptible-Infected-Recovered (SIR) epidemic model, in the sense that the dissemination process and the epidemic thresholds, which achieve maximum coverage with the minimum required transmissions, have been found to be common in certain cases. In this paper, some of these similarities between probabilistic flooding and the SIR epidemic model are identified, particularly with respect to the epidemic thresholds. Both of these epidemic algorithms are experimentally evaluated on a university campus testbed, where a low-cost WSN, consisting of 25 nodes, is deployed. Both algorithm implementations are shown to be efficient at covering a large portion of the network’s nodes, with probabilistic flooding behaving largely in accordance with the considered epidemic thresholds. On the other hand, the implementation of the SIR epidemic model behaves quite unexpectedly, as the epidemic thresholds underestimate sufficient network coverage, a fact that can be attributed to implementation limitations.