Beddington DeAngelis Incidence Research Articles

Dynamics of a cross-superdiffusive SIRS model with delay effects in transmission and treatment.

We investigate the dynamics of a SIRS epidemiological model taking into account cross-superdiffusion and delays in transmission, Beddington-DeAngelis incidence rate and Holling type II treatment. The superdiffusion is induced by inter-country and inter-urban exchange. The linear stability analysis for the steady-state solutions is performed, and the basic reproductive number is calculated. The sensitivity analysis of the basic reproductive number is presented, and we show that some parameters strongly influence the dynamics of the system. A bifurcation analysis to determine the direction and stability of the model is carried out using the normal form and center manifold theorem. The results reveal a proportionality between the transmission delay and the diffusion rate. The numerical results show the formation of patterns in the model, and their epidemiological implications are discussed.

Hybrid stochastic epidemic SIR models with hidden states

This paper focuses on realistic hybrid SIR models that take into account stochasticity. The proposed systems are applicable to most incidence rates that are used in the literature including the bilinear incidence rate, the Beddington–DeAngelis incidence rate, and a Holling type II functional response. Given that many diseases can lead to asymptomatic infections, this paper looks at a system of stochastic differential equations that also includes a class of hidden state individuals, for which the infection status is unknown. Assuming that the direct observation of the percentage of hidden state individuals being infected, α(t), is not given and only a noise-corrupted observation process is available. Using nonlinear filtering techniques in conjunction with an invasion type analysis, this paper shows that the long-term behavior of the disease is governed by a threshold λ∈R that depends on the model parameters. It turns out that if λ<0 the number I(t) of infected individuals converges to zero exponentially fast (extinction). However, if λ>0, the infection is endemic and the system is persistent. We showcase our theorems by applying them in some illuminating examples.

Global Stability for an Endogenous-Reactivated Tuberculosis Model with Beddington-DeAngelis Incidence, Distributed Delay and Relapse.

A tuberculosis (TB) epidemic model with Beddington-DeAngelis incidence and distributed delay is proposed to characterize the interaction between latent period, endogenous reactivation, treatment of latent TB infection, as well as relapse. The basic reproduction number is defined, and the globally asymptotic stability of disease-free equilibrium is shown when , while if the globally asymptotic stability of endemic equilibrium is also acquired. Theoretical results are validated through performing numerical simulations, wherein we detect that TB dynamic behavior between models with discrete and distributed delays could be same and opposite, and TB is more persistent in the model with distributed delay. Besides, increasing the protection level of susceptible and infectious individuals is crucial for the control of TB.

Modeling and stability analysis of cryptosporidiosis transmission dynamics with Beddington-Deangelis incidence

Modeling and stability analysis of cryptosporidiosis transmission dynamics with Beddington-Deangelis incidence

Dynamical complexity of a delay-induced eco-epidemic model with Beddington–DeAngelis incidence rate

Present article has proposed a general eco-epidemic model with disease in predator population subject to Beddington–DeAngelis type incidence rate with the significance of fear in prey individuals and also analyzed the impact of external food supply to the sound predator. As an additional factor in representing the interplay amongst the predator and prey species, the Holling type-II functional response in the context of intra-specific competition within the prey species is taken into consideration. The model dynamics is studied by employing discrete time-delay in prey and gestation delay in predator in order to generate more authentic and natural dynamics. Positivity, uniform boundedness, and uniform persistence of the solutions for the model system have been explained analytically. Furthermore, the extinction criteria of the predator population have been explored and also illustrated by numerical simulations. For the non-delayed model, the existence and stability conditions of all conceivable critical points are investigated associated with the model parameters. The basic fundamental bifurcation assessments of the model reveal the formation of local bifurcations (Hopf-bifurcation and transcritical bifurcation) and provide the parametric region for occurrence of Bautin bifurcation and Gavrilov–Guckenheimer bifurcation. It is also reported that the provision of supplementary food to the sound predator may enable to remove the infection more promptly. Afterward, the stability dynamics of the coexistence state for different configurations of the delay factors have been scrutinized, as well as the delay parameters may produce oscillations via Hopf-bifurcation if they exceed some threshold value. But, when the gestation delay has been incorporated only in infected predators, no critical point can be obtained at which the Hopf-bifurcation takes place for the considered parameter values. Most of the theoretical discoveries have been certified through several numerical experiments constructed with the application of MATLAB and MATCONT.

Fractional order SIR epidemic model with Beddington-De Angelis incidence and Holling type II treatment rate for COVID-19.

In this paper, an attempt has been made to study and investigate a non-linear, non-integer SIR epidemic model for COVID-19 by incorporating Beddington–De Angelis incidence rate and Holling type II saturated cure rate. Beddington–De Angelis incidence rate has been chosen to observe the effects of measure of inhibition taken by both: susceptible and infective. This includes measure of inhibition taken by susceptibles as wearing proper mask, personal hygiene and maintaining social distance and the measure of inhibition taken by infectives may be quarantine or any other available treatment facility. Holling type II treatment rate has been considered for the present model for its ability to capture the effects of available limited treatment facilities in case of Covid 19. To include the neglected effect of memory property in integer order system, Caputo form of non-integer derivative has been considered, which exists in most biological systems. It has been observed that the model is well posed i.e., the solution with a positive initial value is reviewed for non-negativity and boundedness. Basic reproduction number R_{0} is determined by next generation matrix method. Routh Hurwitz criteria has been used to determine the presence and stability of equilibrium points and then stability analyses have been conducted. It has been observed that the disease-free equilibrium Q^{d} is stable for R_{0} < 1 i.e., there will be no infection in the population and the system tends towards the disease-free equilibrium Q^{d} and for R_{0} > 1, it becomes unstable, and the system will tend towards endemic equilibrium Q^{e}. Further, global stability analysis is carried out for both the equilibria using R_{0}. Lastly numerical simulations to assess the effects of various parameters on the dynamics of disease has been performed.

Global dynamics of a delayed HTLV-I infection model with Beddington-DeAngelis incidence and immune impairment

In this paper, we investigate a delayed HTLV-I infection model with Beddington-DeAngelis incidence, CTL immune response and immune impairment. In the model, the intracellular delay and immune response delay are considered. By calculation, we obtain the basic reproductive ratio. By analyzing corresponding characteristic equations, the local stabilities of feasible equilibria are established. By using Lyapunov functionals and LaSalle’s invariance principle, it is shown that when the basic reproductive ratio is less than or equal to unity, the infection-free equilibrium is globally asymptotically stable. When the basic reproductive ratio is greater than unity and immune response delay is equal to zero, the chronic-infection equilibrium is globally asymptotically stable. When the basic reproductive ratio is greater than unity and the intracellular delay is equal to zero, immune response delay will destabilize stability and lead to Hopf bifurcation. Numerical simulations are carried out to illustrate the theoretical results.

Dynamics of a delayed within host model for dengue infection with immune response and Beddington–DeAngelis incidence

In this paper, a new delayed within host model for dengue fever with immune response and Beddington–DeAngelis incidence is investigated. The basic reproduction number is computed. In addition, a detailed analysis on the local and global dynamics of the model is conducted. Finally, sensitivity analysis is carried out on basic reproduction number and numerical simulations are given to elucidate our theoretical results.

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.

Global stability analysis of a delay cell-population model of hepatitis B infection with humoral immune response

In this work, we propose and investigate a delay cell population model of hepatitis B virus (HBV) infection. We suppose spatial diffusion of free HBV particles, and use a Beddington-DeAngelis incidence function to describe viral infection. The model takes into account the exposed hepatocytes and the usually neglected humoral immune response. Moreover, a time delay is introduced to account for the transformation processes necessary for actual HBV production. We naturally find two threshold parameters, namely the basic reproduction number and the humoral immune response reproduction number which completely determine the global stability of the spatially homogeneous equilibria of the model obtained. By constructing appropriate Lyapunov functionals and using LaSalle's invariance principle we show that, if the disease-free equilibrium is globally asymptotically stable. Furthermore, we prove that the endemic equilibrium without humoral immune response and the endemic equilibrium with humoral immune response are globally asymptotically stable if and respectively. Finally, in one dimensional space, we perform some numerical simulations to illustrate the theoretical results obtained.

Global stability for a delayed HIV reactivation model with latent infection and Beddington–DeAngelis incidence

We present a HIV virus-to-cell model with reactivation of latent infection, Beddington–DeAngelis functional response, distributed and discrete time delays. The basic reproduction number R0 is defined, and the global kinetic of the model is studied by characteristic equation and constructing suitable Lyapunov functions. We show that if R0<1 the infection-free equilibrium is globally asymptotically stable, while if R0>1 the endemic equilibrium is globally asymptotically stable.

Viral dynamics of a latent HIV infection model with Beddington-DeAngelis incidence function, B-cell immune response and multiple delays.

In this paper, an HIV infection model with latent infection, Beddington-DeAngelis infection function, B-cell immune response and four time delays is formulated. The well-posedness of the model solution is rigorously derived, and the basic reproduction number $\mathcal{R}_0$ and the B-cell immune response reproduction number $\mathcal{R}_1$ are also obtained. By analyzing the modulus of the characteristic equation and constructing suitable Lyapunov functions, we establish the global asymptotic stability of the uninfected and the B-cell-inactivated equilibria for the four time delays, respectively. Hopf bifurcation occurs at the B-cell-activated equilibrium when the model includes the immune delay, and the B-cell-activated equilibrium is globally asymptotically stable if the model does not include it. Numerical simulations indicate that the increase of the latency delay, the cell infection delay and the virus maturation delay can cause the B-cell-activated equilibrium stabilize, while the increase of the immune delay can cause it destabilize.

Dynamics of a Delayed Epidemic Model with Beddington-Deangelis Incidence Rate and a Constant Infectious Period

Dynamics of a Delayed Epidemic Model with Beddington-Deangelis Incidence Rate and a Constant Infectious Period

RETRACTED: Dynamics of a Delayed Epidemic Model with Beddington-Deangelis Incidence Rate and a Constant Infectious Period

RETRACTED: Dynamics of a Delayed Epidemic Model with Beddington-Deangelis Incidence Rate and a Constant Infectious Period

A nonstandard finite difference scheme and optimal control for an HIV model with Beddington–DeAngelis incidence and cure rate

In this paper, we incorporate the Beddington–DeAngelis incidence rate to a continuous-time HIV infection model with cure rate and a full logistic proliferation rate of $$CD4^+$$ T cells in both uninfected and infected cells. Equilibria and their local stability analysis are discussed. It is shown that the HIV-free equilibrium point is locally asymptotically stable if $${\mathcal {R}}_0<1$$ and unstable if $${\mathcal {R}}_0\ge 1$$ , where $${\mathcal {R}}_0$$ is the basic reproduction number. Whereas, the HIV equilibrium point is locally asymptotically stable if $${\mathcal {R}}_0>1$$ . A nonstandard finite difference method is applied to the continuous model to obtain its discrete counterpart. The scheme applied preserves the main features of the continuous model such as positivity, boundedness of the solutions, equilibria and their local stability. Moreover, an optimal control strategy is applied to the discrete-time model in order to reduce the number of infected cells as well as the number of free HIV particles. Numerical simulations are performed to verify the theoretical analysis obtained.

Stability and Asymptotic Behavior of a Regime-Switching SIRS Model with Beddington–DeAngelis Incidence Rate

A regime-switching SIRS model with Beddington–DeAngelis incidence rate is studied in this paper. First of all, the property that the model we discuss has a unique positive solution is proved and the invariant set is presented. Secondly, by constructing appropriate Lyapunov functionals, global stochastic asymptotic stability of the model under certain conditions is proved. Then, we leave for studying the asymptotic behavior of the model by presenting threshold values and some other conditions for determining disease extinction and persistence. The results show that stochastic noise can inhibit the disease and the behavior will have different phenomena owing to the role of regime-switching. Finally, some examples are given and numerical simulations are presented to confirm our conclusions.

Stability and Hopf bifurcation of a delayed virus infection model with latently infected cells and Beddington–DeAngelis incidence

In this paper, the dynamical behaviors for a five-dimensional virus infection model with Latently Infected Cells and Beddington–DeAngelis incidence are investigated. In the model, four delays which denote the latently infected delay, the intracellular delay, virus production period and CTL response delay are considered. We define the basic reproductive number and the CTL immune reproductive number. By using Lyapunov functionals, LaSalle’s invariance principle and linearization method, the threshold conditions on the stability of each equilibrium are established. It is proved that when the basic reproductive number is less than or equal to unity, the infection-free equilibrium is globally asymptotically stable; when the CTL immune reproductive number is less than or equal to unity and the basic reproductive number is greater than unity, the immune-free infection equilibrium is globally asymptotically stable; when the CTL immune reproductive number is greater than unity and immune response delay is equal to zero, the immune infection equilibrium is globally asymptotically stable. The results show that immune response delay may destabilize the steady state of infection and lead to Hopf bifurcation. The existence of the Hopf bifurcation is discussed by using immune response delay as a bifurcation parameter. Numerical simulations are carried out to justify the analytical results.

Asymptotic properties of a stochastic SIRS epidemic model with nonlinear incidence and varying population sizes

ABSTRACTThis paper studies a class of stochastic SIRS epidemic models with general nonlinear incidence and varying population sizes. A new threshold value is obtained. The sufficient conditions for the global stability of the disease-free equilibrium, the permanence in the mean of the disease and the existence of a unique stationary distribution in probability meaning are established. As applications of the main results, the stochastic SIRS epidemic models with standard incidence, Beddington–DeAngelis incidence and nonlinear incidence rate are discussed, and a series of new criteria for the global stability of the disease-free equilibrium, the permanence in the mean of the disease and the existence of a unique stationary distribution are established.

Stability and Hopf bifurcation analysis for an HIV infection model with Beddington–DeAngelis incidence and two delays

In this paper, we investigate the dynamical properties for a model of delayed differential equations which describes a virus–immune interaction in vivo. The model has two time delays describing time needed for infection of cell and CTLs generation. The model admits three possible equilibria: infection-free equilibrium, CTL-absent infection equilibrium and CTL-present infection equilibrium. The effect of time delay on stability of the equilibria for the CTL immune response model has been studied.

The threshold for a stochastic HIV-1 infection model with Beddington-DeAngelis incidence rate

In this paper, we consider a stochastic HIV-1 infection model with Beddington-DeAngelis incidence rate. Before exploring its long-time behavior we show that there is a global positive solution of this model. Then sufficient conditions for extinction of the disease are established. Moreover, we give sufficient conditions for the existence of a stationary distribution of the model through constructing a suitable stochastic Lyapunov function. The stationary distribution implies that the disease is persistent in the mean. Therefore, a threshold value for the disease to disappear or prevail is obtained. Finally, some numerical examples are illustrated to support our theoretical results.