Search

Aim Coronavirus Disease (COVID‐19) is spreading typically to the human population all over the world and the report suggests that scientists have been trying to map the pattern of the early transmission of Severe Acute Respiratory Syndrome Coronavirus 2 (SARS‐CoV‐2) since it has been reported as an epidemic. Our main aim is to show if the rise‐in‐cases proceeds in a gradual and staggered manner instead of soaring quickly then we can suppress the burden of the health system. In this new case study, we are attempting to show how to control the outbreak of the infectious disease COVID‐19 via mathematical modeling. We have examined that the method of flattening the curve of the coronavirus, which increases the recovery rate of the infected individuals and also helps to decrease the number of deaths. In this pandemic situation, the countries like Russia, India, the United States of America (USA), South Africa, and the United Kingdom (UK) are leading in front where the virus is spreading in an unprecedented way. From our point of view, we establish that if these countries are following the method of flattening the curve like China and South Korea then these countries can also overcome this pandemic situation. Method We propose a Susceptible, Infected, and Recovered (SIR) mathematical model of infectious disease with onset data of COVID‐19 in Wuhan and international cases, which has been propagated in Wuhan City to calculate the transmission rate of the infectious virus COVID‐19 until now. To understand the whole dynamics of the transmission rate of coronavirus, we portray time series diagrams such as growth rate diagram, flattening the pandemic curve diagram, infected and recovered rate diagram, prediction of the transmission of the disease from the available dataset in Wuhan, and internationally exported cases from Wuhan. Results We have observed that the basic reproduction number in Wuhan declined from 2.2 (95% Confidence Interval [CI] 1.15‐4.77) to 1.05 (0.41‐2.39) and the mean incubation period was 5.2 days (95% [CI], 4.1‐7.0). Interestingly the mean value lies between 2 and 2.5 for COVID‐19. The doubling time of COVID‐19 was registered 7.4 days (95% CI, 5.3‐19) in the early stages and now the value decreases to −4.9 days. Similarly, we have observed the doubling time of the epidemic in South Korea decreased to −9.6 days. Currently, the doubling time of the epidemic in Russia, India, and the USA are 19.4 days, 16.4 days, and 41 days, respectively. We have investigated the growth rate of COVID‐19 and plotted the curve flattening diagram against time. Conclusion Via flattening the curve method, China and South Korea control the transmission of the fatal disease COVID‐19 in the human population. Our results show that these two countries initially sustained pandemics in a large portion of the human population in the form of virus outbreaks that basically prevented the virus from spreading further and created ways to prevent community transmission. The majority portion of the people are perfectly fine, who are quarantined strictly and never get sick, but the portion of people who have developed symptoms is quickly isolated further.

In this paper, we have proposed a model system of two preys that mutually contribute to each other with restricted growth rates upon arrival of a generalist predator (who can sexually reproduce) whose predation is followed by Holling type II functional response where three species are self-diffusing for their best of the survival. In spatial case, we investigate the conditions of the blow-up phenomena of sexually reproductive predator of generalist category at finite time. The stability of the system is studied, and we discuss about the conditions of Turing instability. Near bifurcation threshold, major slow down in the dynamical system has been noticed. Variation in the dynamical behaviour of the model system makes structural changes which suggests emergence of patterns. Lastly via numerical simulation, we illustrate different significant patterns such as spots, spots-stripes and stripes which are generated by variation of diffusion parameters and investigate the transition from chaotic to stability properties of the system. Finally, we explain our obtained results where prey exhibits mutual relation among themselves and investigate the populations of the alien/invasive species being fulminated in the ecosystem.

After the works of Lahiri and Banerjee [6] on the idea of relative order (p, q) of entire functions, we introduce in this paper hyper relative order (p, q) of entire functions where p, q are positive integers with p>q and prove sum theorem, product theorem and theorem on derivative.

Species have the general tendency to sustain their own survival chance in the ecology. In this paper, we design different diffusive model systems in which two prey populations make mutualistic relationship in which they are getting benefited from each other with their usual growth rate and sexually reproductive generalist predator preys upon the prey according to their functional responses. In the absence of generalist predator, there is no sign of mutualism between two prey. Sometimes environment may turn favourable for the invasive species, causing the growth of their population to outbreak. Biological control is an adopted strategy to limit harmful populations. To establish a control strategy that decreases the harmful population to healthy levels as opposed to high and risky levels, five different diffusive models have been introduced associated to their functional responses which are either prey dependent (Holling type-III and IV) or predator dependent (Beddington–DeAngelis, Crowley–Martin and and Hassel–Varley). The blow up phenomenon at finite time in spatial cases have been discussed for all the model systems. For each model system, mathematical conditions are established under which species can blow up at finite time. Spatio-temporal dynamics and weakly nonlinear analysis have been thoroughly studied. Stability analysis of these model systems have been investigated and concentrated on discussing the Turing patterns. Theoretical analysis of the patterns formations has been described by using amplitude equations and the results are validated by numerical simulations. Mathematical computations of the structural models have been studied and explored their blow up phenomena under the effects of diffusion with the pattern formation of preys and predator populations in the spatiotemporal domain.The best chance of survival of the species upon diffusion has been discussed elaborately.

The outbreak of coronavirus is spreading at an unprecedented rate to the human populations and taking several thousands of life all over the globe. In this paper, an extension of the well-known susceptible-exposed-infected-recovered (SEIR) family of compartmental model has been introduced with seasonality transmission of SARS-CoV-2. The stability analysis of the coronavirus depends on changing of its basic reproductive ratio. The progress rate of the virus in critical infected cases and the recovery rate have major roles to control this epidemic. Selecting the appropriate critical parameter from the Turing domain, the stability properties of existing patterns is obtained. The outcomes of theoretical studies, which are illustrated via Hopf bifurcation and Turing instabilities, yield the result of numerical simulations around the critical parameter to forecast on controlling this fatal disease. Globally existing solutions of the model has been studied by introducing Tikhonov regularization. The impact of social distancing, lockdown of the country, self-isolation, home quarantine and the wariness of global public health system have significant influence on the parameters of the model system that can alter the effect of recovery rates, mortality rates and active contaminated cases with the progression of time in the real world.

Mutualism is a powerful strategy by which species can assist each other to survive in various harsh environments. These will often take the form of anti-predator strategies. In the current manuscript we consider two preys that mutually benefit each other in the presence of a sexually reproductive generalist predator. In the absence of the predator, the prey grow logistically, and there is no mutualism between them. However, in the presence of the predator the prey will begin to cooperate. We formulate five different models according to their functional responses which are either prey dependent (Holling type-III, IV and Hassel Varley functional responses) or predator–prey dependent (Beddington–DeAngelis and Crowley–Martin functional responses). We perform dynamical analysis of these five models, and also establish the mathematical restrictions under which the species can blow up at a finite time. Interestingly it is observed that the prey populations will blow-up before the predator can. We use our results to comment on a certain paradox in ecological theory, as well as provide further insight into the nature of predator interference and exploding populations of invasive species.