Saturated Incidence Research Articles

Stability and bifurcations in a discrete-time SIVS model with saturated incidence rate

In this paper, a discrete-time SIS epidemic model with vaccination is introduced. The stability of the model is analyzed in the equilibria, after obtaining some basic properties of the model, such as the equilibria, the basic reproduction number, and sufficient conditions for the positivity of solutions. Furthermore, the bifurcations of the model, fold bifurcation, flip bifurcation, and Neimark–Sacker bifurcation are studied. The numerical simulations verify the obtained theoretical results by discussing the diagrams of bifurcations, Lyapunov exponents, and solutions of the model.

Bifurcation and stability of a delayed SIS epidemic model with saturated incidence and treatment rates in heterogeneous networks

In this paper, to characterize the limited availability of medical resources, we incorporate a saturated treatment rate into a network-based susceptible-infected-susceptible (SIS) epidemic model with time delay and nonlinear incidence rate. Analytical study shows the boundedness of solutions, the basic reproduction number R0 and equilibrium points of the proposed system. For any infection delay, we perform both local and global stability analyses for the disease-free equilibrium point by analyzing the characteristic equation and using Lyapunov functional. Furthermore, this system exhibits bifurcation behavior at R0=1 due to the introduction of saturated treatment. More precisely, a backward bifurcation takes place from the disease-free equilibrium point when the saturation constant β is sufficiently large. Under the given conditions, the unique disease-spreading equilibrium point is also proved to be locally asymptotically stable. In addition, we analyze an optimal control problem with consideration of two time-dependent control measures. Several numerical simulations are presented to validate the obtained theoretical results.

Dynamics and simulations of a second order stochastically perturbed SEIQV epidemic model with saturated incidence rate

A second order stochastically perturbed SEIQV epidemic model is studied in this paper. Firstly, we derive sufficient criteria for the existence and uniqueness of an ergodic stationary distribution of positive solutions to the model by establishing a suitable stochastic Lyapunov function. Secondly, we obtain adequate conditions for completing eradication and wiping out of the infectious disease from the basic reproduction number of corresponding deterministic model. Thirdly, distinguished from the previous literatures, the equilibrium stability is investigated by combining Hasminskii theory of stability with Lyapunov function. The new method can also be successfully used for some classical epidemic models. Finally, our analytical results are illustrated by computer simulations, including three examples based on disease originated from reality.

Galerkin time discretization for transmission dynamics of HBV with non-linear saturated incidence rate

This study formulated an autonomous non-linear epidemic model for the numerical solutions of the transmission dynamics of acute and chronic hepatitis B with a saturated incidence rate, which prevailed due to the hepatitis B virus. In this article, a model of HBV is extended by introducing a non-linear saturated incidence rate. That non-linear saturated incidence rate helps to reduce the infected population by improving the immunity of the human body. The continuous Galerkin-Petrove (2) technique a numerical method, is used for finding the solution of the mentioned model. For continuous Galerkin-Petrove (2) technique, we need two variables on every step that must be figured out by solving a square matrix. Different parameters on the population dynamics of susceptible, infected with acute and chronic hepatitis B virus and recovered individuals are analyzed. As it could be seen that the increase in the value of the saturation constant (ξ1) and recovery rate from chronic stage to recovered (ψ2) led to the decrease in the population infected with acute and chronic HBV so susceptible population has large concentration and less population is contaminated and the outcomes are illustrated graphically. The model is solved by utilizing a Runge Kutta method of order (4) scheme and results of both techniques are compared numerically and graphically. Although the numerical and graphical results of both methods (continuous Galerkin-Petrove (2) and Runge Kutta method of order (4)) are the same after comparison the continuous Galerkin-Petrove (2) is more efficient and less time consuming as compared to Runge Kutta method of order (4).

Traveling waves of a differential-difference diffusive Kermack-McKendrick epidemic model with age-structured protection phase

We consider a general class of diffusive Kermack-McKendrick SIR epidemic models with an age-structured protection phase with limited duration, for example due to vaccination or drugs with temporary immunity. A saturated incidence rate is also considered which is more realistic than the bilinear rate. The characteristics method reduces the model to a coupled system of a reaction-diffusion equation and a continuous difference equation with a time-delay and a nonlocal spatial term caused by individuals moving during their protection phase. We study the existence and non-existence of non-trivial traveling wave solutions. We get almost complete information on the threshold and the minimal wave speed that describes the transition between the existence and non-existence of non-trivial traveling waves that indicate whether the epidemic can spread or not. We discuss how model parameters, such as protection rates, affect the minimal wave speed. The difficulty of our model is to combine a reaction-diffusion system with a continuous difference equation. We deal with our problem mainly by using Schauder's fixed point theorem. More precisely, we reduce the problem of the existence of non-trivial traveling wave solutions to the existence of an admissible pair of upper and lower solutions.

Analysis on critical waves for a diffusive epidemic model with saturating incidence rate and infinitely distributed delay

The purpose of this paper is to obtain an existence theorem of critical waves for a diffusive Susceptible‐Infected‐Removed (SIR) epidemic model with saturating incidence rate and infinitely distributed delay. Our results completely answer an aforementioned unsolved questions by using the Schauder's fixed point theorem and constructing a perfect pair of upper‐lower solutions.

Behavioral response of population on transmissibility and saturation incidence of deadly pandemic through fractional order dynamical system

The world entered in another wave of the SARS-CoV-2 due to non-compliance of standard operating procedures appropriately, initiated by respective governments. Apparently, measures like using face masks and social distancing were not observed by populace that ultimately worsens the situation. The behavioral response of the population induces a change in the dynamical outcomes of the pandemic, which is documented in this paper for all intents and purposes. The innovative perception is executed through a compartmental model with the incorporation of fractional calculus and saturation incident rate. In the first instance, the epidemiological model is designed with proportional fractional definition considering the compartmental individuals of susceptible, social distancing, exposed, quarantined, infected, isolated and recovered populations. By virtue of proportional fractional derivative, effective dynamical outcomes of equilibrium states and basic reproduction number are successfully elaborated with memory effect. The expansion of this derivative greatly simplifies the model to integer order while remaining in the fractional context. Subsequently, the memory effects on the asymptotic profiles are demonstrated through various graphical plots and tabulated values. In addition, the inclusion of saturation incident rate further explains the transmissibility of infection for different behavior of susceptible individuals. Mathematically, the results are also validated through comparative analysis of values with the solutions attained from fractional fourth order Runge-Kutta method (FRK4).

Hopf bifurcation of a VEIQS worm propagation model in mobile networks with two delays

The spread of worm virus has brought great loss to our production and life. In this paper, a new Vulnerable-Exposed-Infectious-Quarantined-Secured (VEIQS) worm propagation model with a saturated incidence and two delays is proposed. The local stability of the worm-existence equilibrium and the occurrence of Hopf bifurcation at the critical values of the two delays are obtained by regarding different combinations of time delays as bifurcation parameters. It shows that the model is ideal stable when the time delay is below the critical value and a Hopf bifurcation occurs when the time delay is above the critical value. In particular, direction and stability of the Hopf bifurcation are determined by using the center manifold theorem. Finally, some numerical simulations are presented in order to verify the analytical results.

Dynamics of an SEIRS COVID-19 Epidemic Model with Saturated Incidence and Saturated Treatment Response: Bifurcation Analysis and Simulations

In this work, we study the dynamics of the Coronavirus Disease 2019 pandemic using an SEIRS model with saturated incidence and treatment rates. We derive the basic reproduction number R_0 and study the local stability of the disease-free and endemic states. Since the condition R_0<1 for our model does not determine if the disease will die out, we study the backward bifurcation and Hopf bifurcation to understand the dynamics of the disease at the occurrence of a second wave and the kind of treatment measures needed to curtail it. We present some numerical simulations considering the symptomatic and asymptomatic infections and make a comparison with the reported COVID-19 data for Nigeria. Our results show that the limited availability of medical resources favours the emergence of complex dynamics that complicates the control of the outbreak

The threshold for a stochastic within-host CHIKV virus model with saturated incidence rate

This paper deals with stochastic Chikungunya (CHIKV) virus model along with saturated incidence rate. We show that there exists a unique global positive solution and also we obtain the conditions for the disease to be extinct. We also discuss about the existence of a unique ergodic stationary distribution of the model, through a suitable Lyapunov function. The stationary distribution validates the occurrence of disease; through that, we find the threshold value for prevail and disappear of disease within host. With the help of numerical simulations, we validate the stochastic reproduction number [Formula: see text] as stated in our theoretical findings.

Analisis Kestabilan dan Kontrol Optimal Model Matematika Partisipasi Pemilih pada Pemilihan Umum dengan Saturated Incidence Rate

Voter participation in general elections is an important aspect of a democratic state structure. Participation is determined by the level of public political awareness, if the level of public political awareness is low, voter participation tends to be passive (Abstinence). A mathematical model approach to voter participation in elections that has been modified to a saturated incidence rate is needed to predict voter participation in future elections. This thesis aims to analyze the stability of the equilibrium point and apply the optimal control variable in the form of an awareness campaign. In the model without control variables, we obtain two equilibriums, namely, the non-endemic equilibrium and the endemic equilibrium. Local stability and the existence of endemic equilibrium depend on the basic reproduction number (R0), where R0=bL/(g+m)m. There is voter participation in elections when R0 < 1 and the absence of voter participation in elections when R0 > 1. We also analyze the sensitivity of parameters to determine which parameters are the most influential in this mathematical model. Furthermore, the application of control variables in the mathematical model of voter participation in elections with saturated incidence rate is determined through the Pontryagin Maximum Principle method. Numerical simulation results show that providing control variables in the form of awareness campaign it is quite effective in minimize the number of the voting population who abstained from election.

Stochastic Stability and Analytical Solution with Homotopy Perturbation Method of Multicompartment Non-Linear Epidemic Model with Saturated Rate

In this work, we consider a nonlinear epidemic model with a saturated incidence rate. we consider a population of size N(t) at time t, this population is divided into six subclasses, with N(t)=S(t)+I(t)+I₁(t)+I₂(t)+I₃(t)+Q(t). Where S(t), I(t), I₁(t), I₂(t), I₃(t), and Q(t) denote the sizes of the population susceptible to disease, infectious members, and quarantine members, respectively. We have made the following contributions: 1. The local stabilities of the infection-free equilibrium and endemic equilibrium are; analyzed, respectively. The stability of a disease-free equilibrium and the existence of other nontrivial equilibria can be determined by the ratio called the basic reproductive number. 2. We find the analytical solution of the nonlinear epidemic model by Homotopy perturbation method. 3. Finally the stochastic stabilities. The study of its sections are justified with theorems and demonstrations under certain conditions. In this work, we have used the different references cited in different studies in the three sections already mentioned.

Effects of two patient-specific dosing protocols on measurement of myocardial blood flow with 3D 82Rb cardiac PET

Clinical measurement of myocardial blood flow (MBF) has emerged as an important component of routine PET-CT assessment of myocardial perfusion in patients with known or suspected coronary artery disease. Although multiple society guidelines recommend patient-specific dosing, there is a lack of studies evaluating the efficacy of patient-specific dosing for quantitative MBF accuracy. Two patient-specific dosing protocols (weight- and BMI-adjusted) were retrospectively evaluated in 435 consecutive clinical patients referred for PET myocardial perfusion assessment. MBF was estimated at rest and after regadenoson-induced hyperemia. The effect of dosing protocol on dose reduction, PET scanner saturation, relative perfusion, and image quality was compared. The effect of PET saturation on the accuracy of MBF and myocardial flow reserve (MFR) in remote myocardium was assessed with multivariable linear regression. BMI-adjusted dosing was associated with lower administered 82Rb activities (1036.0 ± 274 vs. 1147 ± 274MBq, p = 0.003) and lower PET scanner saturation incidence (28 vs. 38%, p = 0.006) and severity (median saturation severity index 0.219 ± 0.33 vs. 0.397 ± 0.59%, p = 0.018) compared to weight-adjusted dosing. PET saturation that occurred with either dosing protocol was moderate and resulted in modest remote MBF and MFR biases ranging from 2 to 9% after adjusting for patient age, sex, BMI, rate-pressure product, and LV ejection fraction. No adverse effects of BMI dose adjustment were observed in relative perfusion assessment or image quality. Patient-specific dosing according to BMI is an effective method for guideline-directed dose reduction while maintaining image quality and accuracy for routine MBF and MFR quantification.

Qualitative analysis of a diffusive SIR epidemic model with saturated incidence rate in a heterogeneous environment

In the present paper, we explore a diffusive SIR epidemic model with a saturated incidence rate, logistic growth rate for susceptibles, and Holling type II treatment, in a spatially heterogeneous environment. We establish some uniform bounds of solutions, the global stability of the disease-free equilibrium, uniform persistence, and the existence of endemic equilibria. When the spatial environment is heterogeneous, we determine the asymptotic profile of endemic equilibria if the diffusion rate of the susceptible or infected population is either small or large. Our numerical results show that heterogeneity in the transmission rate enhances the persistence of infection and that the disease may persist even when the basic reproduction number of our model is less than 1, provided that the initial number of infectives is high enough.

Mathematical Modelling and Analysis of Dengue Transmission in Bangladesh with Saturated Incidence Rate and Constant Treatment Function

Dengue is one of the major health problems in Bangladesh and many people are died in recent years due to the severity of this disease. Therefore, in this paper, a SIRS model for the human and SI model for vector population with saturated incidence rate and constant treatment function has been presented to describe the transmission of dengue. The equilibrium points and the basic reproduction number have been computed. The conditions which lead the disease free equilibrium and the endemic equilibrium have been determined. The local stability for the equilibrium points has been established based on the eigenvalues of the Jacobian matrix and the global stability has been analyzed by using the Lyapunov function theory. It is found that the stability of equilibrium points can be controlled by the reproduction number. In order to calculate the infection rate, data for infected human populations have been collected from several health institutions of Bangladesh. Numerical simulations of various compartments have been generated using MATLAB to investigate the influence of the key parameters for the transmission of the disease and to support the analytical results. The effect of treatment function over the infected compartment has been illustrated. The sensitivity of the reproduction number concerning the parameters of the model has been analyzed. Finally, the most sensitive parameter that has the highest effect over reproduction number has been identified.

Global dynamics for a drug-sensitive and drug-resistant mixed strains of HIV infection model with saturated incidence and distributed delays

In this paper, a drug-sensitive and drug-resistant mixed strains of HIV infection model with saturated incidence and distributed infection delays is proposed. The drug-sensitive strain could mutate and become drug-resistant strain during the process of reverse transcription (i.e., SR conversion in short). The nonnegativity and boundedness of solutions are discussed, by introducing two kinds of principle reproduction numbers, i.e., Rs (the reproduction number induced by drug-sensitive strain of HIV virus) and Rr (the reproduction number induced by drug-resistant strain of HIV virus), the existence of equilibria is also acquired. Using the linearization method and constructing the suitable Lyapunov functionals, the criteria are established on the local and global stability of equilibria (including infection-free, drug-sensitive strain, drug-resistant strain and endemic) based on the composite effects of Rs and Rr. We find that there exist competitive exclusion and coexistence phenomenon between drug-sensitive and drug-resistant strains without SR conversion, while uniform persistence between the two kinds of strains with SR conversion. The theoretical results are illustrated by numerical simulations.

Dynamical behavior of a stochastic multigroup staged-progression HIV model with saturated incidence rate and higher-order perturbations

In this paper, we analyze a higher-order stochastically perturbed multigroup staged-progression model for the transmission of HIV with saturated incidence rate. We obtain sufficient conditions for the existence and uniqueness of an ergodic stationary distribution of positive solutions to the system by establishing a suitable stochastic Lyapunov function. In addition, we make up adequate conditions for complete eradication and wiping out the infectious disease. In a biological interpretation, the existence of a stationary distribution implies that the disease will prevail and persist in the long term. Finally, examples and numerical simulations are introduced to validate our theoretical results.

Dynamical behaviour in discrete coupled within-host and between-host epidemic model with environmentally driven and saturation incidence

In this paper, a discrete-time coupled within-host and between-host epidemic model with environmentally driven and saturation incidence is discussed. The model is divided into an isolated fast model and an isolated coupled slow time model by using the way of limit equations. The positivity, boundedness and existence of equilibria for isolated fast and coupled slow time models are investigated, respectively. The global stability of infectious-free and infectious equilibria for the isolated fast time model is established by means of the Lyapunov functions and LaSalle's invariance principle. The local asymptotic stability of disease-free and endemic equilibria for the coupled slow time model is obtained by using the linearization method. When there are two positive equilibria in the coupled slow time model, the existence of backward bifurcation also is established. Furthermore, the correctness of theoretical conclusions and the rationality of conjecture are verified by the numerical simulations.

A fractional order differential equation model for Hepatitis B virus with saturated incidence

This manuscript presents a nonlinear fractional order Hepatitis B virus (HBV) model with saturated incidence. Model governing equations are defined using Caputo fractional derivatives. We use the Adams–Bashforth-Moulton technique to compute numerical solutions of the presented epidemic model. Generally, fractional derivatives are used to model real-world phenomena that contains nonlocal effects, history and/or memory. The main objective is to develop and explore the dynamics of an HBV fractional derivative epidemic model via the Caputo definition. Firstly, we discuss the model basic properties which include positivity of solutions and the invariant regions where the solution set is bounded using the generalized fractional mean value theorem. The model dynamics are described based on the reproduction number R0. Thereafter, we use the fractional Routh-Hurwitz stability criterion as well as simple matrix algebra to determine sufficient conditions for the stability of the equilibrium points. Lastly, we present graphical representation of numerical results. The results indicate the importance of fractional order in modeling HBV transmission dynamics and ensures that by including the memory effects, the model is more appropriate and insightful for such a study.

Threshold of stochastic SIQR epidemic model with saturated incidence

A stochastic SIQR epidemic model with saturated incidence was presented to control the spread of epidemic by taking antiviral prophylaxis and isolation measures. The threshold parameter is identified, which determines the extinction and persistence of the epidemic. The results are illustrated by computer simulations.