A mechanistic model for airborne and direct human-to-human transmission of COVID-19: effect of mitigation strategies and immigration of infectious persons.

The COVID-19 pandemic is the most significant global crisis since World War II that affected almost all the countries of our planet. To control the COVID-19 pandemic outbreak, it is necessary to understand how the virus is transmitted to a susceptible individual and eventually spread in the community. The primary transmission pathway of COVID-19 is human-to-human transmission through infectious droplets. However, a recent study by Greenhalgh et al. (Lancet 397:1603-1605, 2021) demonstrates 10 scientific reasons behind the airborne transmission of SARS-COV-2. In the present study, we introduce a novel mathematical model of COVID-19 that considers the transmission of free viruses in the air beside the transmission of direct contact with an infected person. The basic reproduction number of the epidemic model is calculated using the next-generation operator method and observed that it depends on both the transmission rate of direct contact and free virus contact. The local and global stability of disease-free equilibrium (DFE) is well established. Analytically it is found that there is a forward bifurcation between the DFE and an endemic equilibrium using central manifold theory. Next, we used the nonlinear least-squares technique to identify the best-fitted parameter values in the model from the observed COVID-19 mortality data of two major districts of India. Using estimated parameters for Bangalore urban and Chennai, different control scenarios for mitigation of the disease are investigated. Results indicate that the vaccination of susceptible individuals and treatment of hospitalized patients are very crucial to curtailing the disease in the two locations. It is also found that when a vaccine crisis is there, the public health authorities should prefer to vaccinate the susceptible people compared to the recovered persons who are now healthy. Along with face mask use, treatment of hospitalized patients, and vaccination of susceptibles, immigration should be allowed in a supervised manner so that economy of the overall society remains healthy.

Bifurcations, chaos, and multistability in a nonautonomous predator-prey model with fear.

Classical predator-prey models usually emphasize direct predation as the primary means of interaction between predators and prey. However, several field studies and experiments suggest that the mere presence of predators nearby can reduce prey density by forcing them to adopt costly defensive strategies. Adoption of such kind would cause a substantial change in prey demography. The present paper investigates a predator-prey model in which the predator's consumption rate (described by a functional response) is affected by both prey and predator densities. Perceived fear of predators leads to a drop in prey's birth rate. We also consider both constant and time-varying (seasonal) forms of prey's birth rate and investigate the model system's respective autonomous and nonautonomous implementations. Our analytical studies include finding conditions for the local stability of equilibrium points, the existence, direction of Hopf bifurcation, etc. Numerical illustrations include bifurcation diagrams assisted by phase portraits, construction of isospike and Lyapunov exponent diagrams in bi-parametric space that reveal the rich and complex dynamics embedded in the system. We observe different organized periodic structures within the chaotic regime, multistability between multiple pairs of coexisting attractors with intriguing basins of attractions. Our results show that even relatively slight changes in system parameters, perturbations, or environmental fluctuations may have drastic consequences on population oscillations. Our observations indicate that the fear effect alters the system dynamics significantly and drives an otherwise irregular system toward regularity.

Cooperation delay induced chaos in an ecological system.

In the present paper, we investigate the impact of time delay during cooperative hunting in a predator-prey model. We consider that cooperative predators do not aggregate in a group instantly, but individuals use different stages and strategies such as tactile, visual, vocal cues, or a suitable combination of these to communicate with each other. We observe that delay in hunting cooperation has stabilizing as well as destabilizing effects in the system. Also, for an increase in the strength of the delay, the system dynamics switch multiple times and eventually become chaotic. We see that depending on the threshold of time delay, the system may restore its original state or may go far away from its original state and unable to recollect its memory. Furthermore, we explore the dynamics of the system in different bi-parameter spaces and observe that for a particular range of other parameter values, the system dynamics switch multiple times with an increase of delay in all the planes. Different kinds of multistability behaviors, the coexistence of multiple attractors, and interesting changes in the basins of attraction of the system are also observed. We infer that depending on the initial population size and the strength of cooperation delay, the populations can exhibit stable coexistence, oscillating coexistence, or extinction of the predator species.

Fear effect in prey and hunting cooperation among predators in a Leslie-Gower model.

The predation strategy for predators and the avoidance strategy of prey are important topics in ecology and evolutionary biology. Both prey and predators adjust their behaviours in order to gain the maximal benefits and to increase their biomass for each. In the present paper, we consider a modified Leslie-Gower predator-prey model where predators cooperate during hunting and due to fear of predation risk, prey populations show anti-predator behaviour. We investigate step by step the impact of hunting cooperation and fear effect on the dynamics of the system. We observe that in the absence of fear effect, hunting cooperation can induce both supercritical and subcritical Hopf- bifurcations. It is also observed that fear factor can stabilize the predator-prey system by excluding the existence of periodic solutions and makes the system more robust compared to hunting cooperation. Moreover, the system shows two different types of bi-stabilities behaviour: one is between coexisting equilibrium and limit cycle oscillation, and another is between prey-free equilibrium and coexisting equilibrium. We also observe generalized Hopf-bifurcation and Bogdanov-Takens bifurcation in two parameter bifurcation analysis. We perform extensive numerical simulations for supporting evidence of our analytical findings.