Backward bifurcation, oscillations and chaos in an eco-epidemiological model with fear effect

An eco-epidemiological model with the impact of fear

In this study, we propose and analyze an eco-epidemiological model with disease in prey and incorporated the effect of fear on prey species due to predator population. We assume that the prey population grows logistically in the absence of predator species, and the disease is limited to the prey population only. We divide the total prey population into two distinct classes: susceptible prey and infected prey. Predator populations are not infected by the diseases, though feed both the susceptible and infected prey. Due to the fear of predators, the prey population becomes more vigilant and moves away from suspected predators. Such a foraging activity of prey reduces the chance of infection among susceptible prey by lowering the contact with infected prey. We assume that the fear of predators has no effect on infected prey as they are more vigilant. Positivity, boundedness, and uniform persistence of the proposed model are investigated. The biologically feasible equilibrium points and their stability are analyzed. We establish the conditions for the Hopf bifurcation of the proposed model around the endemic steady state. As the level of fear increases, the system moves toward the steady state from a limit cycle oscillation. The increasing level of fear cannot wipe out the diseases from the system, but the amplitude of the infected prey decreases as the level of fear is increased. The system changes its stability as the rate of infection increases, and the predator becomes extinct when the rate of infection in prey is high enough though predators are not infected by the disease.

A delay non-autonomous model for the combined effects of fear, prey refuge and additional food for predator

In this paper, we investigate the combined effects of fear, prey refuge and additional food for predator in a predator-prey system with Beddington type functional response. We observe oscillatory behaviour of the system in the absence of fear, refuge and additional food whereas the system shows stable dynamics if anyone of these three factors is introduced. After analysing the behaviour of system with fear, refuge and additional food, we find that the system destabilizes due to fear factor whereas refuge and additional food stabilize the system by killing persistent oscillations. We extend our model by considering the fact that after sensing the chemical/vocal cue, prey takes some time for assessing the predation risk. The delayed system shows chaotic dynamics through multiple stability switches for increasing values of time delay. Moreover, we see the impact of seasonal change in the level of fear on the delayed as well as non-delayed system.

Dynamics of a stage-structured predator-prey model: cost and benefit of fear-induced group defense

In the predator-prey system, predators can affect the prey population (1) by direct killing and (2) by inducing predation fear, which ultimately force preys to adopt some anti-predator strategies. However, the anti-predator strategy is not the same for all individual preys of different life stages. Also, anti-predator behavior has both cost and benefit, but most of the mathematical models observed the dynamics by incorporating its cost only. In the present study, we formulate a predator-prey model dividing the prey population into two stages: juvenile and adult. We assume that adult preys are only adapting group defense as an anti-predator strategy when they are sensitive to predation. Group defense plays a positive role for adult prey by reducing their predation, but, on the negative side, it simultaneously decreases their reproductive potential. A parameter, anti-predator sensitivity is introduced to interlink both the benefit and cost of group defense. Our result shows that when adult preys are not showing anti-predator behavior, with an increase of maturation rate, the system exhibits a population cycle of abruptly increasing amplitude, which may drive all species of the system to extinction. Anti-predator sensitivity may exclude oscillation through homoclinic bifurcation and avert the prey population for any possible random extinction. Anti-predator sensitivity also decreases the predator population density and produces bistable dynamics. Higher values of anti-predator sensitivity may lead to the extinction of the predator population and benefit adult preys to persist with large population density. Below a threshold value of anti-predator sensitivity, it may possible to retain the predator population in the system by increasing the fear level of the predator. We also observe our fear-induced stage-structured model exhibits interesting and rich dynamical behaviors, various types of bistabilities in different bi-parameter planes. Finally, we discuss the potential impact of our findings.

The impact of fear factor and self-defence on the dynamics of predator-prey model with digestion delay

In this paper, we propose both deterministic and stochastic predator-prey models with digestion delay, incorporating fear factor and self-defence. For the deterministic model, the existence and stability of the equilibrium are investigated and the occurrence of Hopf bifurcation is studied. For the stochastic model, we investigate the existence of a unique global positive solution of the model and analyze the asymptotic behavior of the global solution around the equilibriums of the deterministic model. Finally, numerical simulations are carried out to verify our analytical results, which indicate that the intensity of white noise, fear factor and self-defence have a significant relationship with the dynamics of the predator-prey model and expand the theoretical analyses.

A NONAUTONOMOUS MODEL FOR THE INTERACTIVE EFFECTS OF FEAR, REFUGE AND ADDITIONAL FOOD IN A PREY–PREDATOR SYSTEM

The importance of fear, refuge and additional food is being increasingly recognized in recent studies, but their combined effects need to be explored. In this paper, we investigate the joint effects of these three ecologically important factors in a prey–predator system with Crowly–Martin type functional response. We find that the fear of predator significantly affects the densities of prey and predator populations whereas the presence of prey refuge and additional food for predator are recognized to have potential impacts to sustain prey and predator in the habitat, respectively. The fear of predator induces limit cycle oscillations while an oscillatory system becomes stable on increasing the refuge. The system first undergoes a supercritical Hopf-bifurcation and then a subcritical Hopf-bifurcation on increasing either the growth rate of prey or growth rate of predator due to additional food. Increasing the quality/quantity of additional food after a certain value causes extinction of prey species and rapid incline in the predator population. An extension is made in the model by considering the seasonal variations in the cost of fear of predator, prey refuge and growth rate of predator due to additional food. The nonautonomous model is shown to exhibit globally attractive positive periodic solution. Moreover, complex dynamics such as higher periodic solutions and bursting patterns are observed due to seasonal variations in the rate parameters.

Dynamic Panel Estimate-Based Health Surveillance of SARS-CoV-2 Infection Rates to Inform Public Health Policy: Model Development and Validation

BACKGROUND

SARS-CoV-2, the novel coronavirus that causes COVID-19, is a global pandemic with higher mortality and morbidity than any other virus in the last 100 years. Without public health surveillance, policy makers cannot know where and how the disease is accelerating, decelerating, and shifting. Unfortunately, existing models of COVID-19 contagion rely on parameters such as the basic reproduction number and use static statistical methods that do not capture all the relevant dynamics needed for surveillance. Existing surveillance methods use data that are subject to significant measurement error and other contaminants.

OBJECTIVE

The aim of this study is to provide a proof of concept of the creation of surveillance metrics that correct for measurement error and data contamination to determine when it is safe to ease pandemic restrictions. We applied state-of-the-art statistical modeling to existing internet data to derive the best available estimates of the state-level dynamics of COVID-19 infection in the United States.

METHODS

Dynamic panel data (DPD) models were estimated with the Arellano-Bond estimator using the generalized method of moments. This statistical technique enables control of various deficiencies in a data set. The validity of the model and statistical technique was tested.

RESULTS

A Wald chi-square test of the explanatory power of the statistical approach indicated that it is valid (χ210=1489.84, P<.001), and a Sargan chi-square test indicated that the model identification is valid (χ2946=935.52, P=.59). The 7-day persistence rate for the week of June 27 to July 3 was 0.5188 (P<.001), meaning that every 10,000 new cases in the prior week were associated with 5188 cases 7 days later. For the week of July 4 to 10, the 7-day persistence rate increased by 0.2691 (P=.003), indicating that every 10,000 new cases in the prior week were associated with 7879 new cases 7 days later. Applied to the reported number of cases, these results indicate an increase of almost 100 additional new cases per day per state for the week of July 4-10. This signifies an increase in the reproduction parameter in the contagion models and corroborates the hypothesis that economic reopening without applying best public health practices is associated with a resurgence of the pandemic.

CONCLUSIONS

DPD models successfully correct for measurement error and data contamination and are useful to derive surveillance metrics. The opening of America involves two certainties: the country will be COVID-19-free only when there is an effective vaccine, and the "social" end of the pandemic will occur before the "medical" end. Therefore, improved surveillance metrics are needed to inform leaders of how to open sections of the United States more safely. DPD models can inform this reopening in combination with the extraction of COVID-19 data from existing websites.