SIQR Model Research Articles

Matrix measures, stability and contraction theory for dynamical systems on time scales

<p style='text-indent:20px;'>This paper is concerned with the study of the stability of dynamical systems evolving on time scales. We first formalize the notion of matrix measures on time scales, prove some of their key properties and make use of this notion to study both linear and nonlinear dynamical systems on time scales. Specifically, we start with considering linear time-varying systems and, for these, we prove a time scale analogous of an upper bound due to Coppel. We make use of this upper bound to give stability and input-to-state stability conditions for linear time-varying systems. Then, we consider nonlinear time-varying dynamical systems on time scales and establish a sufficient condition for the convergence of the solutions. Finally, after linking our results to the existence of a Lyapunov function, we make use of our approach to study certain epidemic dynamics and complex networks. For the former, we give a sufficient condition on the parameters of a SIQR model on time scales ensuring that its solutions converge to the disease-free solution. For the latter, we first give a sufficient condition for pinning synchronization of complex time scale networks and then use this condition to study certain collective opinion dynamics. The theoretical results are complemented with simulations.</p>

Dynamics of SIQR epidemic model with fractional order derivative.

The dynamics of COVID-19 (Coronavirus Disease-2019) transmission are described using a fractional order SIQR model. The stability analysis of the model is performed. To obtain semi-analytic solutions to the model, the Iterative Laplace Transform Method [ILTM] is implemented. Real-time data from COVID-19 cases in India and Brazil is employed to estimate the parameters of the fractional order SIQR model. Numerical solutions obtained using Adam–Bashforth–Moultonpredictor–corrector technique is compared with those obtained by ILTM. It is observed that the fractional order of the derivatives is more effective in studying the dynamics of the spread of COVID-19 in comparison to integral order of the SIQR model.

Numerical and bifurcation analysis of SIQR model

In this mathematical research paper, we analyze in detail the basic SIQR epidemic model. We calculate its reproductive value R0, equilibrium points and analyze the stability of the SIQR system by using Routh-Hurwitz criterion in detail. We also find out the bifurcation value of the SIQR epidemic model by using Routh-Hurwitz criterion. Also, SIQR system is solved numerically by using four different mathematical techniques that are forward Euler scheme, Runge-Kutta (RK-4) method, variational iteration method and nonstandard finite difference scheme (NSFD). Analytical and graphical calculations show that the NSFD method preserves all the important conditions of the basic SIQR epidemic model while the rest three techniques fail to preserve the essential conditions of the system. Convergence analysis of the NSFD scheme has also been performed.

Asymptotic properties of a stochastic SIQR epidemic model with Lévy Jumps and Beddington-DeAngelis incidence rate

In this paper, we propose a stochastic SIQR model and discuss the impact of Lévy jumps and Beddington-DeAngelis incidence rate on the transmission of diseases. We prove that our proposed model admits a unique global positive solution and an invariant positive set. We establish sufficient conditions for the extinction and persistence of the disease in the population using some stochastic calculus background. We illustrate our theoretical results by numerical simulations. We infer that the white and Lévy noises influence the transmission dynamic of the system.

Stationary Distribution and Extinction in a Stochastic SIQR Epidemic Model Incorporating Media Coverage and Markovian Switching

This paper is concerned with the dynamic characteristics of the SIQR model with media coverage and regime switching. Firstly, the existence of the unique positive solution of the proposed system is investigated. Secondly, by constructing a suitable random Lyapunov function, some sufficient conditions for the existence of a stationary distribution is obtained. Meanwhile, the conditions for extinction is also given. Finally, some numerical simulation examples are carried out to demonstrate the effectiveness of theoretical results.

An Analysis of CTMC Stochastic Models with Quarantine on the Spread of Tuberculosis Diseases

The SIQRS epidemic model developed in this study is intended to analyze the spread characteristics of the infectious disease tuberculosis. It is a modification of the SIQR model developed by Cao et al., using a stochastic model called the Continuous Time Markov Chains (CTMC) approach. Further analysis of the SIQRS model was done to determine the transitional probability, the outbreak probability, the expected time until disease extinction and to simulate the effect of quarantine treatment on the expected time until disease extinction. Based on the simulation it can be concluded that a decrease of the healing rate together with an increase of the transmission rate changes the basic reproduction number (R0), the expected number of infected individuals (m), the time until disease extinction, and the outbreak probability. A disease outbreak will occur if both R0>1 and m>1 hold. Also, based on the simulation it was concluded that the decrease of the healing rate and the increase of the transmission rate cause increases of R0 and m. An increase of the quarantine rate reduces the expected time to disease extinction, R0 and m. As a consequence, the disease will gradually disappear from the system.

Modeling School Reopening Options in Response to COVID-19

School administrators have the daunting challenge of rolling out school reopening policies and safety standards. While it is important to have children in school to maximize learning, it is critical to ensure that it is done in a manner which prioritizes the safety of staff and students on campus. This research explores models that can be used to examine the spread of COVID-19, and then applies them to evaluate school reopening policies within the United States. A variety of factors are accounted for, including the student population, the rate at which students are infected, the rate at which students are quarantined, and the proportion of students that are susceptible to COVID-19. We primarily utilize Euler’s method and nondimensionalization of the differential equations for each model. Our graphical results are displayed using Excel. We show that multiple variables such as rate of infection and recovery can be written solely as a function of one parameter: R0. We first analyze the SIR model and then introduce the new scenario of quarantining students to derive the SIQR model. Our findings show that reducing and maintaining R0 below 1 is the key to reopening schools and we outline practices to help achieve this outcome.

Self-organized wavy infection curve of COVID-19

Exploiting the SIQR model for COVID-19, I show that the wavy infection curve in Japan is the result of fluctuation of policy on isolation measure imposed by the government and obeyed by citizens. Assuming the infection coefficient be a two-valued function of the number of daily confirmed new cases, I show that when the removal rate of infected individuals is between these two values, the wavy infection curve is self-organized. On the basis of the infection curve, I classify the outbreak of COVID-19 into five types and show that these differences can be related to the relative magnitude of the transmission coefficient and the quarantine rate of infected individuals.

A study of SIQR model with Holling type–II incidence rate

A study of SIQR model with Holling type–II incidence rate

Mathematical analysis of an age structured epidemic model with a quarantine class

In this paper, an age structured epidemic Susceptible-Infected-Quarantined-Recovered-Infected (SIQRI) model is proposed, where we will focus on the role of individuals that leave the R-class before being completely recovered and thus will participate again to the disease transmission. We investigate the asymptotic behavior of solutions by studying the stability of both trivial and positive equilibria. In order to see the impact of the different model parameters like the relapse rate on the qualitative behavior of our system, we firstly, give an explicit expression of the basic reproduction number R0, which is a combination of the classical basic reproduction number for the SIQR model and some other model parameters, corresponding to the individuals infected by the relapsed ones. It will be shown that, if R0 ≤ 1, the disease free equilibrium is globally asymptotically stable and becomes unstable for R0 > 1. Secondly, while R0 > 1, a suitable Lyapunov functional is constructed to prove that the unique endemic equilibrium is globally asymptotically stable on some subset Ω0.

Mathematical modeling and dynamic analysis of SIQR model with delay for pandemic COVID-19.

On the basis of the SIQR epidemic model, we consider the impact of treatment time on the epidemic situation, and we present a differential equation model with time-delay according to the characteristics of COVID-19. Firstly, we analyze the existence and stability of the equilibria in the modified COVID-19 epidemic model. Secondly, we analyze the existence of Hopf bifurcation, and derive the normal form of Hopf bifurcation by using the multiple time scales method. Then, we determine the direction of Hopf bifurcation and the stability of bifurcating periodic solutions. Finally, we carry out numerical simulations to verify the correctness of theoretical analysis with actual parameters, and show conclusions associated with the critical treatment time and the effect on epidemic for treatment time.

Exact properties of SIQR model for COVID-19

The SIQR model is reformulated where compartments for infected and quarantined are redefined so as to be appropriate to COVID-19, and exact properties of the model are presented. It is shown that the maximum number of infected at large depends strongly on the quarantine rate and that the quarantine measure is more effective than the lockdown measure in controlling the pandemic. The peak of the number of quarantined patients is shown to appear some time later than the time that the number of infected becomes maximum. On the basis of the expected utility theory, a theoretical framework to find out an optimum strategy in the space of lockdown measure and quarantine measure is proposed for minimizing the maximum number of infected and for controlling the outbreak of pandemic at its early stage.

Bifurcation analysis of a fractional-order SIQR model with double time delays

In this paper, a fractional-order delayed SIQR model with nonlinear incidence rate is investigated. Two time delays are incorporated in the model to describe the incubation period and the time caused by the healing cycle. By analyzing the associated characteristic equations, the stability of the endemic equilibrium and the existence of Hopf bifurcation are obtained in three different cases. Besides, the critical values of time delays at which a Hopf bifurcation occurs are obtained, and the influence of the fractional order on the dynamics behavior of the system is also investigated. Numerically, it has been shown that when the endemic equilibrium is locally stable, the convergence rate of the system becomes slower with the increase of the fractional order. Besides, our studies also imply that the decline of the fractional order may convert a oscillatory system into a stable one. Furthermore, we find in all these three cases, the bifurcation values are very sensitive to the change of the fractional order, and they decrease with the increase of the order, which means the Hopf bifurcation gradually occurs in advance.

Age-group-targeted testing for COVID-19 as a new prevention strategy.

Robust testing and tracing are key to fighting the menace of coronavirus disease 2019 (COVID-19). This outbreak has progressed with tremendous impact on human life, society and economy. In this paper, we propose an age-structured SIQR model to track the progression of the pandemic in India, Italy and USA, taking into account the different age structures of these countries. We have made predictions about the disease dynamics, identified the most infected age groups and analysed the effectiveness of social distancing measures taken in the early stages of infection. The basic reproductive ratio {mathfrak {R}}_0 has been numerically calculated for each country. We propose a strategy of age-targeted testing, with increased testing in the most proportionally infected age groups. We observe a marked flattening of the infection curve upon simulating increased testing in the 15–40 year age groups in India. Thus, we conclude that social distancing and widespread testing are effective methods of control, with emphasis on testing and identifying the hot spots of highly infected populations. It has also been suggested that a complete lockdown, followed by lockdowns in selected regions, is more effective than the reverse.

Long-Time Behavior of a Stochastic SIQR Model with Markov Switching

In this paper, we consider a stochastic SIQR model with a perturbed disease transmission coefficient. We determine the threshold λ that is used to classify the extinction and permanence of the model. Concretely, λ 0, the model is strongly stochastically permanent. Our findings are considered significant improvements over the results in Liu et al. (Appl. Math. Comput. 316, 310–325 2018).

Analysis of the outbreak of COVID-19 in Japan by SIQR model.

The SIQR model is exploited to analyze the outbreak of COVID-19 in Japan where the number of the daily confirmed new cases is explicitly treated as an observable. It is assumed that the society consists of four compartments; susceptible individuals (S), infected individuals at large (I), quarantined patients (Q) and recovered individuals (R), and the time evolution of the pandemic is described by a set of ordinary differential equations. It is shown that the quarantine rate can be determined from the time dependence of the daily confirmed new cases, from which the number of infected individuals can be estimated. The infection rate and quarantine rate are determined for the period from mid-February to mid-April in Japan and transmission characteristics of the initial stages of the outbreak in Japan are analyzed in connection with the policies employed by the government. The effectiveness of different measures is discussed for controlling the outbreak and it is shown that identifying patients through PCR (Polymerase Chain Reaction) testing and isolating them in a quarantine is more effective than lockdown measures aimed at inhibiting social interactions of the general population. An effective reproduction number for infected individuals at large is introduced which is appropriate to epidemics controlled by quarantine measures.

Stationary Distribution and Extinction of a Stochastic SIQR Model with Saturated Incidence Rate

In this paper, we consider a stochastic SIQR epidemic model with saturated incidence rate. By constructing a proper Lyapunov function, we obtain the existence and uniqueness of positive solution for this SIQR model. Furthermore, we study the dynamical properties of this stochastic SIQR model; that is, (i) we establish the sufficient condition for the existence of ergodic stationary distribution of the model; (ii) we obtain the extinction of the disease under some conditions. At last, numerical simulations are introduced to illustrate our theoretical results.

A nonautonomous epidemic model on time scales

ABSTRACTWe obtain conditions for permanence and extinction of the infection for a nonautonomous SIQR model defined on an arbitrary time scale. The threshold conditions are given by some numbers that play the role of the basic reproduction number in this setting. As a particular case of our result, we recover several threshold conditions obtained in the literature, on discrete or continuous time, for autonomous, periodic models and general nonautonomous models and we also discuss some new situations, including an aperiodic time scale.

Analysis of an SIQR model

Analysis of an SIQR model

A SIQR model with a nonlinear incidence

A SIQR model with a nonlinear incidence