SIR Epidemic Model Research Articles

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.

Propagation dynamics of a reaction–diffusion two-group SIR epidemic model with constant recruitment

In this paper, we focus on asymptotic speeds of spread for a reaction–diffusion two-group SIR epidemic model with constant recruitment, which lacks the comparison principle. More precisely, if [Formula: see text], then the solution of the system converges to the disease-free equilibrium as [Formula: see text] and if [Formula: see text], there exists a critical speed [Formula: see text] such that the solution of the system is uniformly persistent with [Formula: see text] and the infection dies out with [Formula: see text] for any [Formula: see text]. Finally, some numerical experiments are presented to modeling the propagation dynamics of the system.

A new acceleration of variational iteration method for initial value problems

This paper is devoted to accelerating the variational iteration method (VIM) to solve a nonlinear initial value problem. For this purpose, the redundant calculations of the conventional VIM are removed, and the complex integration is evaluated recursively and quickly with a suitable numerical approximation. The convergence of the proposed method is proven, and an efficient implementation algorithm is presented. This method is successfully applied to solve three applied equations, including the Riccati equation, the Lane–Emden equation, and the SIR epidemic model with a constant vaccination rate. Numerical simulations show that this improvement has significantly increased the method’s speed and enlarged its convergence region.

Phase portraits of an SIR epidemic model

In this paper, we classify the phase portraits of an SIR epidemic dynamics model. Depending on the values of the parameters, this model can exhibit seven different phase portraits. In particular, from a biological point of view we prove that the unique attractors of this model are one or two equilibrium points depending on the values of the parameters and from the phase portraits follow the basins of attraction of these equilibria.

Dynamics of nonlocal and local SIR diffusive epidemic model with free boundaries

This paper is concerned with nonlocal and local diffusive SIR epidemic model with free boundaries including convolution, which is natural extension of reaction diffusion systems with free boundary problems and local diffusions. The existence of unique global solution for this model is considered. Dichotomy of the spreading and vanishing is established. A spreading barrier line is found to determine whether the spreading of disease will fail finally. The spreading of disease will fail when it cannot spread across the spreading barrier line l∗, while it will be successful when it transcends over this barrier line. The results show that if the basic reproduction number R0<1, the spreading of disease will fail eventually, and if R0>1+d2μ2+α, the spreading of disease will get success finally. We also find that the spreading coefficients play important role in the spreading achievement. When 1+βθμ1<R0<1+d2μ2+α, the spreading coefficient decides whether the spreading of disease will be successful. It is shown that the spreading will be successful when the spreading coefficient is relatively big, while the spreading will fail if the spreading coefficient is relatively small.

Bifurcation of an SIRS Model with a Modified Nonlinear Incidence Rate

An SIRS epidemic model with a modified nonlinear incidence rate is studied, which describes that the infectivity is strong at first as the emergence of a new disease or the reemergence of an old disease, but then the psychological effect will weaken the infectivity. Lastly, the infectivity goes to a saturation state as a result of a crowding effect. The nonlinearity of the functional form of the incidence of infection is modified, which is more reasonable biologically. We analyze the stability of the associated equilibria, and the basic reproduction number and the critical value which determine the dynamics of the model are derived. The bifurcation analysis is presented, including backward bifurcation, saddle-node bifurcation, Bogdanov–Takens bifurcation of codimension two and Hopf bifurcation. To study Hopf bifurcation of codimension three of the model when some assumptions hold, the focus values are calculated. Numerical simulations are shown to verify our results.

Traveling wave solutions in a nonlocal dispersal SIR epidemic model with nonlocal time-delay and general nonlinear incidences

This paper investigates a nonlocal dispersal epidemic model under the multiple nonlocal distributed delays and nonlinear incidence effects. First, the minimal wave speed [Formula: see text] and the basic reproduction number [Formula: see text] are defined, which determine the existence of traveling wave solutions. Second, with the help of the upper and lower solutions, Schauder’s fixed point theorem, and limiting techniques, the traveling waves satisfying some asymptotic boundary conditions are discussed. Specifically, when [Formula: see text], for every speed [Formula: see text] there exists a traveling wave solution satisfying the boundary conditions, and there is no such traveling wave solution for any [Formula: see text] when [Formula: see text] or [Formula: see text] when [Formula: see text]. Finally, we analyze the effects of nonlocal time delay on the minimum wave speed.

Big data in COVID-19 prevention and control: Modeling and analysis report

The COVID-19 epidemic has brought great external impact to China. China is facing complex internal and external environmental challenges. SIR epidemic model is a classical partition model, which is widely used to predict the progress of COVID-19. Although the SIR model may be useful in simulating multiple epidemics, it may not be sufficient to describe the spread of COVID-19. Therefore, some modifications were made and used to study the spread and control of COVID-19 epidemic on the SIR model of COVID-19 disease. Expand it by increasing the link between tracking and other interventions. By studying the SEIR model considering the interaction between human and infectious source. In this paper, we will use the classical SIR model to simulate and predict the spread of COVID-19. By distinguishing between confirmed and undiagnosed individuals, the development of COVID-19 is characterized by phased changes. Based on the preliminary data analysis of the epidemic on various industries, the actual impact of the epidemic on society was quantitatively analyzed.

Dynamical analysis of a discrete-time SIR epidemic model

In this paper, a discrete-time seasonally forced SIR epidemic model is investigated for different types of bifurcations. Although, many researchers already suggested numerically that this model can exhibit chaotic dynamics but not much focus is given to the bifurcation theory of the model. We prove analytically and numerically the existence of different types of bifurcations in the model. First, the one and two parameters bifurcations of this model are investigated by computing their critical normal form coefficients. Secondly, the flip, Neimark–Sacker, and strong resonances bifurcations are detected for this model. The critical coefficients identify the scenario associated with each bifurcation. The complete complex dynamical behavior of the model is investigated. Some graphical representations of the model are presented to verify the obtained results.

A nonlocal diffusion SIR epidemic model with nonlocal incidence rate and free boundaries

This paper is concerned with the spreading or vanishing of an epidemic disease which is characterized by a nonlocal diffusion SIR model with nonlocal incidence rate and double free boundaries. We prove that the disease will vanish if the basic reproduction number , or the initial area , the initial datum , and the expanding ability are sufficiently small when . Otherwise, the disease will spread to the whole area if , when is suitably large or is small but is large enough. Moreover, we also show the long‐time asymptotic limit of the solution when vanishing happens.

SIRS epidemic modeling using fractional-ordered differential equations: Role of fear effect

In this paper, an SIRS epidemic model using Grunwald–Letnikov fractional-order derivative is formulated with the help of a nonlinear system of fractional differential equations to analyze the effects of fear in the population during the outbreak of deadly infectious diseases. The criteria for the spread or extinction of the disease are derived and discussed on the basis of the basic reproduction number. The condition for the existence of Hopf bifurcation is discussed considering fractional order as a bifurcation parameter. Additionally, using the Grunwald–Letnikov approximation, the simulation is carried out to confirm the validity of analytic results graphically. Using the real data of COVID-19 in India recorded during the second wave from 15 May 2021 to 15 December 2021, we estimate the model parameters and find that the fractional-order model gives the closer forecast of the disease than the classical one. Both the analytical results and numerical simulations presented in this study suggest different policies for controlling or eradicating many infectious diseases.

COVID-19 SIR model: Bifurcation analysis and optimal control

This paper proposes a SIR epidemic model with vital dynamics to control or eliminate the spread of the COVID-19 epidemic considering the constant population, saturated treatment, and direct–indirect transmission rate of the model. We demonstrate positivity, boundness and calculate the disease-free equilibrium point and basic reproduction number from the model. We use the Jacobian matrix and the Lyapunov function to analyze the local and global stability, respectively. It is observed that indirect infection increases the basic reproduction number and gives rise to multiple endemic diseases. We perform transcritical, forward, backward, and Hopf bifurcation analyses. We propose two control parameters (Use of face mask, hand sanitizer, social distancing, and vaccination) to minimize the spread of the coronavirus. We use Pontryagin’s maximum principle to solve the optimal control problem and demonstrate the results numerically.

Computational aspects of an epidemic model involving stochastic partial differential equations

This paper deals with the study of the reaction–diffusion epidemic model perturbed with time noise. It has various applications such as disease in population models of humans, wildlife, and many others. The stochastic SIR model is numerically investigated with the proposed stochastic backward Euler scheme and proposed stochastic implicit finite difference (IFD) scheme. The stability of the proposed methods is shown with Von Neumann criteria and both schemes are unconditionally stable. Both schemes are consistent with systems of the equations in the mean square sense. The numerical solution obtained by the proposed stochastic backward Euler scheme and solutions converges towards an equilibrium but it has negative and divergent behavior for some values. The numerical solution gained by the proposed IFD scheme preserves the positivity and also solutions converge towards endemic and disease-free equilibrium. We have used two problems to check our findings. The graphical behavior of the stochastic SIR model is much adjacent to the classical SIR epidemic model when noise strength approaches zero. The three-dimensional plots of the susceptible and infected individuals are drawn for two cases of endemic equilibrium and disease-free equilibriums. The results show the efficacy of the proposed stochastic IFD scheme.

Final Size for Epidemic Models with Asymptomatic Transmission

The final infection size is defined as the total number of individuals that become infected throughout an epidemic. Despite its importance for predicting the fraction of the population that will end infected, it does not capture which part of the infected population will present symptoms. Knowing this information is relevant because it is related to the severity of the epidemics. The objective of this work is to give a formula for the total number of symptomatic cases throughout an epidemic. Specifically, we focus on different types of structured SIR epidemic models (in which infected individuals can possibly become symptomatic before recovering), and we compute the accumulated number of symptomatic cases when time goes to infinity using a probabilistic approach. The methodology behind the strategy we follow is relatively independent of the details of the model.

Stability analysis of an SIR model with alert class modified saturated incidence rate and Holling functional type-II treatment

Abstract This study discusses an SIR epidemic model with modified saturated incidence rates and Holling functional type-II therapy. In this study, we take the new alert compartment (A) in the SIR compartment model. Consider the modified non-linear incidence rate from the susceptible to the infected class and the second non-linear incidence rate from the alert to the infected class. Further, we investigate the elementary reproduction number, the equilibrium points of the model, and their stability. We apply manifold theory to discuss bifurcations of the disease-free equilibrium point. This study shows that the infected population decreases with the Holling functional type II treatment rate. It also shows that the number of infected people decreases when the psychological rate increases and the contact rate decreases.

Asymptotic Behavior of Delayed SIR Epidemic Models of COVID-19 with Diffusion

Asymptotic Behavior of Delayed SIR Epidemic Models of COVID-19 with Diffusion

Lévy noise with infinite activity and the impact on the dynamic of an SIRS epidemic model

Lévy noise with infinite activity and the impact on the dynamic of an SIRS epidemic model

Modeling the influence of the information domain on countermeasure effectiveness in case of COVID-19

A common way to model an epidemic — restricted to contagion aspects only — is a modification of the Kermack-McKendrick SIR Epidemic model (SIR model) with differential equations. (Mis-)Information about epidemics may influence the behavior of the people and thus the course of epidemics as well. We have thus coupled an extended SIR model of the COVID-19 pandemic with a compartment model of the (mis-)information-based attitude of the population towards epidemic countermeasures. The resulting combined model is checked concerning basic plausibility properties like positivity and boundedness. It is calibrated using COVID-19 data from RKI and attitude data provided by the COVID-19 Snapshot Monitoring (COSMO) study. The values of parameters without corresponding observation data have been determined using an L2 -fit under mild additional assumptions. The predictions of the calibrated model are essentially in accordance with observations. An uncertainty analysis of the model shows, that our results are in principle stable under measurement errors. We also assessed the scale, at which specific parameters can influence the evolution of epidemics. Another result of the paper is that in a multi-domain epidemic model, the notion of controlled reproduction number has to be redefined when being used as an indicator of the future evolution of epidemics.

Editorial for the special issue “Control‐theoretic approaches for systems in the life sciences”

Editorial for the special issue “Control<scp>‐theoretic</scp> approaches for systems in the life sciences”