Deterministic and stochastic analysis of an eco-epidemiological model

Bifurcation and onset of chaos in an eco-epidemiological system with the influence of time delay

In the present article, we investigated a delay-based eco-epidemic prey-predator system in the presence of environmental fluctuations where predators engage with susceptible and infected prey, adopting Holling type II and ratio-dependent functional responses, respectively. During the study of the considered model, we identify each potential equilibrium point and its local stability criterion. The basic reproduction number has been computed, and the backward bifurcation about the disease-free equilibrium point was analyzed. The article illustrates Hopf bifurcation, global stability at the endemic equilibrium point, and their graphical depiction. We look over the variations in the dynamics of non-delay, delayed, and stochastic systems, revealing that a fixed level of temporal delay results in chaotic motion for the increasing strength of the saturation constant yet is potentially controlled by the predator growth rate. To study the dynamic behavior of the solution of the considered system and verify all theoretical results, we use numerical simulation and minutely analyze the influence of model parameters on the solution of the considered system. The stochastic transition is studied by varying the strength of stochastic fluctuation and the effect of delay.

The effect of noise in an HIV infection model with cytotoxic T-lymphocyte impairment

The human immunodeficiency virus (HIV) interacts with the immune cells within the human body, where the environment is uncertain and noisy. Stochastic models can successfully encapsulate the effect of such a noisy environment compared to their deterministic counterparts. The human immune system is complex but well-coordinated with various immune cells like C D ⁴T cells, dendritic cells, and cytotoxic T-lymphocyte (CTL) cells, among many others. The CTL can kill the antigenic cells after its recognition. However, the efficacy of CTL in removing the infected C D ⁴T cells is progressively compromised in HIV-infected individuals. This paper considers a noise-induced HIV-immune cell interaction model with immune impairment. A multiplicative white noise is introduced in the infection rate parameter to represent the fluctuations around the average value of the rate parameter as a causative effect of the noise. We analyzed the deterministic and stochastic models and prescribed sufficient conditions for infection eradication and persistence. It is determined under what parametric restrictions the asymptotic solutions of the noise-induced system will be a limiting case of the deterministic solutions. Simulation results revealed that the solutions of the deterministic system either converge to a CTL-dominated interior equilibrium or a CTL-free immunodeficient equilibrium, depending on the initial values of the system. Stochastic analysis divulged that higher noise might be helpful in the infection removal process. The extinction time of infected C D ⁴T cells for some fixed immune impairment gradually decreases with increasing noise intensity and follows the power law.

Deterministic and Stochastic Prey–Predator Model for Three Predators and a Single Prey

In this paper, a deterministic prey–predator model is proposed and analyzed. The interaction between three predators and a single prey was investigated. The impact of harvesting on the three predators was studied, and we concluded that the dynamics of the population can be controlled by harvesting. Some sufficient conditions were obtained to ensure the local and global stability of equilibrium points. The transcritical bifurcation was investigated using Sotomayor’s theorem. We performed a stochastic extension of the deterministic model to study the fluctuation environmental factors. The existence of a unique global positive solution for the stochastic model was investigated. The exponential–mean–squared stability of the resulting stochastic differential equation model was examined, and it was found to be dependent on the harvesting effort. Theoretical results are illustrated using numerical simulations.

CHRONIC WASTING DISEASE MODELING: AN OVERVIEW

Chronic wasting disease (CWD) is an infectious and fatal prion disease occurring in the family Cervidae. To update the research community regarding the status quo of CWD epidemic models, we conducted a meta-analysis on CWD research. We collected data from peer-reviewed articles published since 1980, when CWD was first diagnosed, until December 2018. We explored the analytical methods used historically to understand CWD. We used 14 standardized variables to assess overall analytical approaches of CWD research communities, data used, and the modeling methods used. We found that CWD modeling initiated in the early 2000s and has increased since then. Connectivity of the research community was heavily reliant on a cluster of CWD researchers. Studies focused primarily on regression and compartment-based models, population-level approaches, and host species of game management concern. Similarly, CWD research focused on single populations, species, and locations, neglecting modeling using community ecology and biogeographic approaches. Chronic wasting disease detection relied on classic diagnostic methods with limited sensitivity for most stages of infection. Overall, we found that past modeling efforts generated a solid baseline for understanding CWD in wildlife and increased our knowledge on infectious prion ecology. Future analytical efforts should consider more sensitive diagnostic methods to quantify uncertainty and broader scale studies to elucidate CWD transmission beyond population-level approaches. Considering that infectious prions may not follow biological rules of well-known wildlife pathogens (i.e., viruses, bacteria, fungi), assumptions used when modeling other infectious disease may not apply for CWD. Chronic wasting disease is a new challenge in wildlife epidemiology.

Stability and bifurcation analysis of a diffusive modified Leslie-Gower prey-predator model with prey infection and Beddington DeAngelis functional response

In this paper, we present and analyze a spatio-temporal eco-epidemiological model of a prey predator system where prey population is infected with a disease. The prey population is divided into two categories, susceptible and infected. The susceptible prey is assumed to grow logistically in the absence of disease and predation. The predator population follows the modified Leslie-Gower dynamics and predates both the susceptible and infected prey population with Beddington-DeAngelis and Holling type II functional responses, respectively. The boundedness of solutions, existence and stability conditions of the biologically feasible equilibrium points of the system both in the absence and presence of diffusion are discussed. It is found that the disease can be eradicated if the rate of transmission of the disease is less than the death rate of the infected prey. The system undergoes a transcritical and pitchfork bifurcation at the Disease Free Equilibrium Point when the prey infection rate crosses a certain threshold value. Hopf bifurcation analysis is also carried out in the absence of diffusion, which shows the existence of periodic solution of the system around the Disease Free Equilibrium Point and the Endemic Equilibrium Point when the ratio of the rate of intrinsic growth rate of predator to prey crosses a certain threshold value. The system remains locally asymptotically stable in the presence of diffusion around the disease free equilibrium point once it is locally asymptotically stable in the absence of diffusion. The Analytical results show that the effect of diffusion can be managed by appropriately choosing conditions on the parameters of the local interaction of the system. Numerical simulations are carried out to validate our analytical findings.