Saturated Incidence Rate Research Articles

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.

An SEIR model with modified saturated incidence rate and Holling type II treatment function

Abstract In this article, the behavior of an susceptible exposed infected recovered (SEIR) epidemic model with nonlinear incidence rate and Holling type II treatment function is presented and analyzed. Reproduction number of the model is calculated. Equilibrium points are determined. Disease-free equilibrium exists when R0 is below 1. Behavior of disease-free equilibrium is examined at R0 = 1. Endemic equilibrium exists when R0 crosses 1. Stability of both equilibrium points is investigated locally and globally. Simulation is provided to support the result.

A Lyapunov functional for vaccination model on the dynamics of cholera epidemic with non-linear incidence of infection

A new deterministic susceptible-exposed-infectious-vaccinated-removed-pathogen (SEIVRB) cholera epidemic model with combined mass action incidence and saturated incidence rates is proposed and analyzed. A threshold level of vaccine coverage necessary for controlling or eradicating cholera has been determined and analyzed using the next generation matrix approach. It is shown that the higher values of vaccine coverage that are lower than the threshold value significantly reduces the number of infected individuals whenever basic reproduction number is less than unity, and the cholera would persist in the populations whenever the model basic reproduction number exceeds unity. The global stability for cholera free equilibrium state and cholera endemic equilibrium state of the model system is investigated using Lyapunov functional approach and Lasalle invariance principle, which are found to be globally asymptotically stable at both equilibrium states. Numerical simulations and graphical illustrations is presented to support the analytical results found in the study.

Bifurcation analysis of an SIS epidemic model with a generalized non-monotonic and saturated incidence rate

In this paper, a new generalized non-monotonic and saturated incidence rate was introduced into a susceptible-infected-susceptible (SIS) epidemic model to account for inhibitory effect and crowding effect. The dynamic properties of the model were studied by qualitative theory and bifurcation theory. It is shown that when the influence of psychological factors is large, the model has only disease-free equilibrium point, and this disease-free equilibrium point is globally asymptotically stable; when the influence of psychological factors is small, for some parameter conditions, the model has a unique endemic equilibrium point, which is a cusp point of co-dimension two, and for other parameter conditions the model has two endemic equilibrium points, one of which could be weak focus or center. In addition, the results of the model undergoing saddle-node bifurcation, Hopf bifurcation and Bogdanov–Takens bifurcation as the parameters vary were also proved. These results shed light on the impact of psychological behavior of susceptible people on the disease transmission.

Stationary distribution and probability density function of a stochastic waterborne pathogen model with saturated direct and indirect transmissions

We investigate the dynamical behavior of deterministic and stochastic waterborne pathogen models with saturated incidence rates. In particular, the indirect transmission via person–water–person contact is half‐saturated and the direct transmission by person–person contact takes the saturated form. Possible equilibrium points of the model are investigated, and their stability criterion is discussed. Basic reproduction number of the model is obtained through the next‐generation matrix method. It has been shown that the disease‐free equilibrium is locally stable when and unstable for . Furthermore, a stochastic Lyapunov method is adopted to establish sufficient conditions for the existence and uniqueness of an ergodic stationary distribution of the proposed stochastic model, which reveals that the infection will persist if . We also obtain an exact expression of the probability density function near the quasi‐endemic equilibrium of stochastic system. Finally, numerical simulations are carried out to support our analytical findings. Results suggest that saturation constants in both direct and indirect transmissions play a positive role in controlling disease. Our results may provide some new insights for the elimination of waterborne disease.

Effect of Delay in SMS Worm Propagation in Mobile Network with Saturated Incidence Rate

Effect of Delay in SMS Worm Propagation in Mobile Network with Saturated Incidence Rate

Modelling disease spread with spatio-temporal fractional derivative equations and saturated incidence rate.

The global analysis of a spatio-temporal fractional order SIR infection model with saturated incidence function is suggested and studied in this paper. The dynamics of the infection is described by three partial differential equations including a time-fractional derivative order for each one of them. The equations of our model describe the evolution of the susceptible, the infected and the recovered individuals with taking into account the spatial diffusion for each compartment. We will choose a saturated incidence rate in order to describe the nonlinear force of the infection. First, we will prove the well-posedness of our suggested model in terms of existence and uniqueness of the solution. Also in this context, the boundedness and the positivity of solutions are established. Afterward, we will give the forms of the disease-free equilibrium and the endemic one. It was demonstrated that the global stability of the each equilibrium depends mainly on the basic reproduction number. Finally, numerical simulations are performed to validate the theoretical results and to show the effect of vaccination in reducing the infection severity. It was shown that the fractional derivative order has no effect on the equilibria stability but only on the convergence speed towards the steady states. It was also observed that vaccination is amongst the good strategies in controlling the disease spread.

Modeling COVID-19 Using a Modified SVIR Compartmental Model and LSTM-Estimated Parameters

This article presents a modified version of the SVIR compartmental model for predicting the evolution of the COVID-19 pandemic, which incorporates vaccination and a saturated incidence rate, as well as piece-wise time-dependent parameters that enable self-regulation based on the epidemic trend. We have established the positivity of the ODE version of the model and explored its local stability. Artificial neural networks are used to estimate time-dependent parameters. Numerical simulations are conducted using a fourth-order Runge–Kutta numerical scheme, and the results are compared and validated against actual data from the Autonomous Communities of Spain. The modified model also includes explicit parameters to examine potential future scenarios. In addition, the modified SVIR model is transformed into a system of one-dimensional PDEs with diffusive terms, and solved using a finite volume framework with fifth-order WENO reconstruction in space and an RK3-TVD scheme for time integration. Overall, this work demonstrates the effectiveness of the modified SVIR model and its potential for improving our understanding of the COVID-19 pandemic and supporting decision-making in public health.

The stability analysis of a co-circulation model for COVID-19, dengue, and zika with nonlinear incidence rates and vaccination strategies

This paper aims to study the impacts of COVID-19 and dengue vaccinations on the dynamics of zika transmission by developing a vaccination model with the incorporation of saturated incidence rates. Analyses are performed to assess the qualitative behavior of the model. Carrying out bifurcation analysis of the model, it was concluded that co-infection, super-infection and also re-infection with same or different disease could trigger backward bifurcation. Employing well-formulated Lyapunov functions, the model’s equilibria are shown to be globally stable for a certain scenario. Moreover, global sensitivity analyses are performed out to assess the impact of dominant parameters that drive each disease’s dynamics and its co-infection. Model fitting is performed on the actual data for the state of Amazonas in Brazil. The fittings reveal that our model behaves very well with the data. The significance of saturated incidence rates on the dynamics of three diseases is also highlighted. Based on the numerical investigation of the model, it was observed that increased vaccination efforts against COVID-19 and dengue could positively impact zika dynamics and the co-spread of triple infections.

A Well-Posed Fractional Order Cholera Model with Saturated Incidence Rate.

A fractional-order cholera model in the Caputo sense is constructed. The model is an extension of the Susceptible-Infected-Recovered (SIR) epidemic model. The transmission dynamics of the disease are studied by incorporating the saturated incidence rate into the model. This is particularly important since assuming that the increase in incidence for a large number of infected individualsis equivalent to a small number of infected individualsdoes not make much sense. The positivity, boundedness, existence, and uniqueness of the solution of the model are also studied. Equilibrium solutions are computed, and their stability analyses are shown to depend on a threshold quantity, the basic reproduction ratio (R0). It is clearly shown that if R0<1, the disease-free equilibrium is locally asymptotically stable, whereas if R0>1, the endemic equilibrium exists and is locally asymptotically stable. Numerical simulations are carried out to support the analytic results and to show the significance of the fractional order from the biological point of view. Furthermore, the significance of awareness is studied in the numerical section.

Dynamical behavior of a fractional order SIR model with stability analysis

The fractional order SIR model with a Holling type II saturated incidence rate and treatment rate are explored in this manuscript in the Caputo order fractional derivative approach. The existence and uniqueness criterion, as well as non-negativity and boundedness of the solution of the new model have been established. The stability analysis of the model shows that the system is locally as well as globally asymptotically stable at disease-free equilibrium point E0 when R0<1 and at epidemic equilibrium E∗ when R0>1. For R0=1 at E0 the model exhibits a forward bifurcation. Fractional-order Taylor’s approach is utilized to approximate the solution of the proposed model. Graphical demonstrations and numerical simulations have been presented using MATLAB.

Mathematical Modeling of Foot and Mouth Disease Spread on Livestock using Saturated Incidence Rate

Foot and Mouth Disease (FMD) is an acute infectious disease that attacks livestock, thus threatening the availability of food and the husbandry industry. This paper discusses the formulation of a mathematical model for the spread of FMD in livestock with a saturated incidence rate. The research method used is quantitative mathematical modeling with simulation, with stages including problem identification, determining assumptions, model formulation, analysis and model simulation. The discussion results obtained two equilibrium points, namely the non-endemic equilibrium point and the endemic equilibrium point, and then analyzed for stability. Numerical simulation is presented using Runge-Kutta approximation with MATLAB. Furthermore, after a sensitivity analysis, the parameters that greatly influenced the spread of FMD were direct or indirect contact (which led to the entry of the FMD virus) and the supporting capacity of livestock. Then the most influential parameter in reducing the spread of FMD is the application of culling on exposed animals and infected animals. The FMD modeling is a form of mathematical application to simulate the spread of disease on livestock.

Dynamic analysis and optimal control of worm propagation model with saturated incidence rate

In order to prevent the propagation of computer worms effectively, based on the latent character of worms, the exposed compartments of computer and USB device are introduced respectively, and a series of computer worm propagation models with saturation incidence rate are proposed. The qualitative behavior of the proposed model is studied. Firstly, the threshold R0 of the model is derived by using the next-generation matrix method, which completely characterized the stability of disease free equilibrium and endemic equilibrium. If R0 &amp;lt; 1, the disease free equilibrium is asymptotically stable, implying that the worm dies out eventually and its attack remains under control; if R0 &amp;gt; 1, the asymptotic stability of endemic equilibrium under certain conditions is proved, which means that the worm is always persistent and uncontrollable under such conditions. Secondly, the theoretical results are verified by numerical study, in which the relative importance of each parameter in worm prevalence is evaluated by sensitivity analysis. Finally, so as to minimize the number of computer and USB device carrying computer worms in short span of time, the worm propagation model is extended to incorporate three control strategies. The Pontryagin’s maximum principle is used to characterize the controls’ optimal levels. According to the control effect diagram, the combined strategy is effective in minimizing the transmission dynamics of worm virus in computer and USB devices populations respectively.

Analysis on a diffusive SIRS epidemic model with logistic source and saturated incidence rate

This paper provides the qualitative analysis on a diffusive SIRS epidemic model with logistic source and saturated incidence rate. We first establish the uniform boundedness of the parabolic system, and then explore the extinction/persistence of the infectious disease via the basic reproduction number. The effects of logistic source, dispersal, saturation, and spatial heterogeneity on asymptotic behaviors of the endemic equilibrium are also investigated. Our study show that the disease can not be controlled by limiting the dispersal rate and large saturation may cause the elimination of disease.

Dynamics of a fractional-order rubella disease model with vertical transmission and saturated incidence rate

Dynamics of a fractional-order rubella disease model with vertical transmission and saturated incidence rate

Sensitivity analysis of cassava mosaic disease with saturation incidence rate model

&lt;abstract&gt;&lt;p&gt;Cassava mosaic disease (CMD) is caused by a virus transmitted by the whitefly. This disease can destroy cassava at any stage of its growth and it resulted in lower cassava yields. In this paper, we developed a mathematical model for the epidemic of cassava mosaic disease with a deterministic model which has saturation incidence rates. This model aims to explain the effect of vectors on cassava disease outbreaks. First, this model was analyzed using standard dynamic methods to determine the behavior of the solution. We found the existence and condition of disease-free and endemic steady state. The basic reproductive number ($ R_0 $) is obtained by using the next-generation method which $ R_0 $ helps assess the ability to spread infectious diseases. Second, the stability of the steady state was analyzed, then we obtain the condition of existence of local stability and global stability at each steady state of this model. Third, analysis of the sensitivity indices in the threshold number to determine the effect of the various parameters. Finally, the results of the theoretical model were validated by numerical simulations. It is represented by various graphs converging at a steady state and stable.&lt;/p&gt;&lt;/abstract&gt;

Modeling and analysis of a stochastic giving-up-smoking model with quit smoking duration.

Smoking has gradually become a very common behavior, and the related situation in different groups also presents different forms. Due to the differences of individual smoking cessation time and the interference of environmental factors in the spread of smoking behavior, we establish a stochastic giving up smoking model with quit-smoking duration. We also consider the saturated incidence rate. The total population is composed of potential smokers, smokers, quitters and removed. By using Itô's formula and constructing appropriate Lyapunov functions, we first ensure the existence of a unique global positive solution of the stochastic model. In addition, a threshold condition for extinction and permanence of smoking behavior is deduced. If the intensity of white noise is small, and $ \widetilde{\mathcal{R}}_0 < 1 $, smokers will eventually become extinct. If $ \widetilde{\mathcal{R}}_0 > 1 $, smoking will last. Then, the sufficient condition for the existence of a unique stationary distribution of the smoking phenomenon is studied as $ R_0^s > 1 $. Finally, conclusions are explained by numerical simulations.

Impact of general incidence function on three-strain SEIAR model.

We investigate the behavior of a complex three-strain model with a generalized incidence rate. The incidence rate is an essential aspect of the model as it determines the number of new infections emerging. The mathematical model comprises thirteen nonlinear ordinary differential equations with susceptible, exposed, symptomatic, asymptomatic and recovered compartments. The model is well-posed and verified through existence, positivity and boundedness. Eight equilibria comprise a disease-free equilibria and seven endemic equilibrium points following the existence of three strains. The basic reproduction numbers $ \mathfrak{R}_{01} $, $ \mathfrak{R}_{02} $ and $ \mathfrak{R}_{03} $ represent the dominance of strain 1, strain 2 and strain 3 in the environment for new strain emergence. The model establishes local stability at a disease-free equilibrium point. Numerical simulations endorse the impact of general incidence rates, including bi-linear, saturated, Beddington DeAngelis, non-monotone and Crowley Martin incidence rates.

Dynamics and simulations of stochastic COVID-19 epidemic model using Legendre spectral collocation method

&lt;abstract&gt;&lt;p&gt;The aim of this study is to investigate the dynamics of epidemic transmission of COVID-19 SEIR stochastic model with generalized saturated incidence rate. We assume that the random perturbations depends on white noises, which implies that it is directly proportional to the steady states. The existence and uniqueness of the positive solution along with the stability analysis is provided under disease-free and endemic equilibrium conditions for asymptotically stable transmission dynamics of the model. An epidemiological metric based on the ratio of basic reproduction is used to describe the transmission of an infectious disease using different parameters values involve in the proposed model. A higher order scheme based on Legendre spectral collocation method is used for the numerical simulations. For the better understanding of the proposed scheme, a comparison is made with the deterministic counterpart. In order to confirm the theoretical analysis, we provide a number of numerical examples.&lt;/p&gt;&lt;/abstract&gt;

Stochastic stability and impulsive vaccination of multicompartment nonlinear epidemic model with incidence rate

In this work, we consider a multicompartment nonlinear epidemic model with temporary immunity and a saturated incidence rate. N(t) at time t, this population is divide into seven sub-classes. N(t) = S(t) + E(t) + I(t) + I1(t) + I2(t) + I3(t) + Q(t). where S(t),E(t); I(t); I(t); I1(t),I2(t); I3(t) and Q(t) denote the sizes of the population susceptible to disease, exposed, infectious members and quarantine members with the possibility of infection through temporary immunity, respectively.The stability of a disease-free status equilibrium and the existence of endemic equilibrium determined by the ratio called the basic reproductive number. The multicompartment non linear epidemic model with saturated rate has been studied the stochastic stability of the free disease equilibrium under certain conditions, and obtain the conditions of global attractivity of the infection.