Spreading speed for a delayed epidemic model incorporating Beddington–DeAngelis incidence rate

The biphasic decay of blood viraemia in patients being treated for human immunodeficiency virus type 1 (HIV-1) infection has been explained as the decay of two distinct populations of cells: the rapid death of productively infected cells followed by the much slower elimination of a second population the identity of which remains unknown. Here we advance an alternative explanation based on the immune response against a single population of infected cells. We show that the biphasic decay can be explained simply, without invoking multiple compartments: viral load falls quickly while cytotoxic T lymphocytes (CTL) are still abundant, and more slowly as CTL disappear. We propose a method to test this idea, and develop a framework that is readily applicable to treatment of other infections.

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.

Hybrid stochastic epidemic SIR models with hidden states

This study introduces an epidemic model with a Beddington-DeAngelis-type incidence rate and Holling type II treatment rate. The Beddington-DeAngelis incidence rate is used to evaluate the effectiveness of inhibitory measures implemented by susceptible and infected individuals. Moreover, the choice of Holling type II treatment rate in our model aims to assess the impact of limited treatment facilities in the context of disease outbreaks. First, the well-posed nature of the model is analyzed, and then, we further investigated the local and global stability analysis along with bifurcation of co-dimensions 1 (transcritical, Hopf, saddle-node) and 2 (Bogdanov-Takens, generalized Hopf) for the system. Moreover, we incorporate a time-delayed model to investigate the effect of incubation delay on disease transmission. We provide a rigorous demonstration of the existence of chaos and establish the conditions that lead to chaotic dynamics and chaos control. Additionally, sensitivity analysis is performed using partial rank correlation coefficient and extended Fourier amplitude sensitivity test methods. Furthermore, we delve into optimal control strategies using Pontryagin's maximum principle and assess the influence of delays in state and control parameters on model dynamics. Again, a stochastic epidemic model is formulated and analyzed using a continuous-time Markov chain model for infectious propagation. Analytical estimation of the likelihood of disease extinction and the occurrence of an epidemic is conducted using the branching process approximation. The spatial system presents a comprehensive stability analysis and yielding criteria for Turing instability. Moreover, we have generated the noise-induced pattern to assess the effect of white noise in the populations. Additionally, a case study has been conducted to estimate the model parameters, utilizing COVID-19 data from Poland and HIV/AIDS data from India. Finally, all theoretical results are validated through numerical simulations. This article extensively explores various modeling techniques, like deterministic, stochastic, statistical, pattern formation(noise-induced), model fitting, and other modeling perspectives, highlighting the significance of the inhibitory effects exerted by susceptible and infected populations.

In this paper, the stochastic SIR epidemic model with Beddington–DeAngelis incidence rate is investigated. We classify the model by introducing a threshold value \(\lambda \). To be more specific, we show that if \(\lambda 0\). In this case, we derive that the model under consideration has a unique invariant probability measure. We also depict the support of invariant probability measure and prove the convergence in total variation norm of transition probabilities to the invariant measure. Some of numerical examples are given to illustrate our results.

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.

Stationary distribution and extinction of SIR model with nonlinear incident rate under Markovian switching

Global stability for an HIV-1 infection model with Beddington–DeAngelis incidence rate and CTL immune response

We estimate the spreading speeds in diffusive epidemic models with nonlocal delays, nonlinear incidence rate and constant recruitment rate. The purpose is to model the process that the infective invades the habitat of the susceptible, and they coexist eventually. In order to focus on our idea, a system with a nonlinear incidence rate is firstly studied, which implies a saturation level of the infective individuals and monotone incidence rate. When the initial value of the infective has nonempty compact support, we prove the rough spreading speed that equals the minimal wave speed of traveling wave solutions in the known results. Then for a general (nonmonotone) incidence rate, we obtain the spreading speeds by constructing auxiliary systems admitting a monotone incidence rate, and prove the convergence of solutions on any compact spatial interval. Furthermore, some numerical examples are given to estimate the invasion speed and show the nontrivial effect of time delay and spatial nonlocality, which implies that the stronger spatial nonlocality leads to larger spreading speeds.

In this paper, we study sufficient conditions for the permanence and ergodicity of a stochastic susceptible-infected-recovered (SIR) epidemic model with Beddington-DeAngelis incidence rate in both of non-degenerate and degenerate cases. The conditions obtained in fact are close to the necessary one. We also characterize the support of the invariant probability measure and prove the convergence in total variation norm of the transition probability to the invariant measure. Some of numerical examples are given to illustrate our results.

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

This paper discuss a modified mathematical model of Zika virus transmission and analyzes the impact of the awareness programs on social media the modification of of Zika Virus model with saturated incidence rate. The Beddington-De Angelis functional responses used to describe the interaction between a suspected human and an infected human. The dynamics of the model were analyzed by identifying the disease-free (DFE) and endemic equilibrium (END). Next Generation Matrix (NGM) was used to determine the Basic Reproduction Number. The stability of DFE and END were analyzed locally by computing the determinant of Jacobian. The DFE was identified as locally stable when the basic reproduction number was less than unity; and was identified as unstable otherwise. Meanwhile, the END was identified as existents when the basic reproduction number was greater than unity. The Routh-Hurwitz Criterion showed that the END was locally stable under a specific condition. A sensitivity analysis was also computed to determine the most influential parameter value of the model. In the end, the stability of DFE and END were also identified numerically depending on certain parameter values.

In this paper, a humoral and cellular immunity virus dynamics model with the Beddington‐DeAngelis incidence rate is set up. We derive the basic reproductive number R0, the cytotoxic T lymphocytes immune response reproductive number R1, the humoral immune response reproductive number R2, humoral immune response competitive reproductive number R3, and cytotoxic T lymphocytes immune response competitive reproductive number R4, and a full description of the relation between the existence of the equilibria and reproductive numbers is given. The global properties of the five equilibria are obtained by constructing Lyapunov functions. Copyright © 2014 John Wiley &amp; Sons, Ltd.

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

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 {mathcal {R}}_0 is defined, and the globally asymptotic stability of disease-free equilibrium is shown when {mathcal {R}}_0<1, while if {mathcal {R}}_0>1 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.