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In this research paper, we consider a Leslie–Gower Reaction–Diffusion (RD) model with a predator-driven Allee term in the prey population. We derive conditions for the existence of nontrivial solutions, uniform boundedness, local stability at co-existing equilibrium points, and Hopf bifurcation criteria from the temporal system. We identify sufficient conditions for Turing instability with no-flux boundary condition for the spatial system. Our investigation delves into the analysis of diffusion-induced Turing instability, incorporating stability conditions for the constant steady-state in the spatial model. We also investigate the conditions for the existence and nonexistence of nonconstant steady states in the diffusion-induced model. During numerical simulations, we observe that the predator-driven Allee term is essential for the model to generate Turing structures. Our findings reveal intriguing properties within the RD system, demonstrating its ability to produce patterns within the Turing domain. The simulation confirms that cold–hot spots and stripes-like patterns (a mixture of spots and strips) arises for different strengths of the predation parameter and Allee parameter. In contrast, we observe that for the above threshold value of the Allee parameter, the above-mentioned patterns may disappear from the system. Interestingly, we also observe that the stationary system produces patterns for both large and small amplitudes of perturbation in the vicinity of the Turing boundary. Our research may contribute valuable insights into the Allee effect and enhance our understanding of predator–prey interactions in naturalistic environments. [ABSTRACT FROM AUTHOR]

In the present study we have considered a predator-prey model with disease in both population. We have also considered the harvesting in all species and allee in the susceptible prey species. We have worked out the conditions for which the equilibrium points exists and local stability of equilibrium points. We have derived the conditions for persistence of all species. Our numerical simulation results reveals the global behaviors of our system. Disease in prey species accelerates the system to produce limit cycle oscillations and disease in predator species stabilizes the oscillatory system. harvesting in the species also stabilized the oscillatory system. Harvesting in the both species can remove the disease from both population. If we increase the values of allee parameter we have observed the exchange of stability of some important equilibrium points. [ABSTRACT FROM AUTHOR]

Several reports in India indicate hospitals and quarantined centers are COVID‐19 hotspots. To study the transmission occurring from the hospitals and as well as from the community, we developed a mechanistic model with a lockdown effect. Using daily COVID‐19 cases data from six states and overall India, we estimated several important parameters of our model. Moreover, we provided an estimation of the effective (RT), the basic (R0), the community (RC), and the hospital (RH) reproduction numbers. We forecast COVID‐19 notified cases from May 3, 2020, till May 20, 2020, under five different lockdown scenarios in the seven locations. Our analysis suggests that 65% to 99% of the new COVID‐19 cases are currently asymptomatic in those locations. Besides, about 1–16% of the total COVID‐19 transmission are currently occurring from hospital‐based contact and these percentage can increase up to 69% in some locations. Furthermore, the hospital‐based transmission rate (β2) has significant positive (0.65 to 0.8) and negative (‐0.58 to ‐0.23) correlation with R0 and the effectiveness of lockdown, respectively. Therefore, a much larger COVID‐19 outbreak may trigger from the hospital‐based transmission. In most of the locations, model forecast from May 3, 2020, till May 20, 2020, indicates a two‐times increase in cumulative cases in comparison to total observed cases up to April 29, 2020. Based on our results, we proposed a containment policy that may reduce the threat of a larger COVID‐19 outbreak in the future. [ABSTRACT FROM AUTHOR]

We propose and analyze an eco-epidemic model with disease in predator. The model dytamics is studied with gestation delay in predator and incubation delay in disease transmission along with four different incidence functions. Our findings re-establish the claim of de Jong et al. that the nass action and standard incidence functions behave in a similar fashion. In the absence of timeielay, the stability conditions of the equilibrium points are derived in terms of basic reproduction numbers. We observe that disease has a stabilization effect. Further, we study the stability dynamics of the interior equilibrium for various combinations of the delay factors and observe that the delay may produce oscillations through a Hopf bifurcation. The stability and direction of Hopf bifurcation are determined by applying the normal form theory and the center manifold theorem. Our analytical results are illustrated by numerical simulations. [ABSTRACT FROM AUTHOR]

Shukla et al. [Appl. Math. Comp. 196(2) 782-790 (2008)], have proposed and analyzed a nonlinear model for the algal bloom in a lake caused by excessive flow of nutrients from domestic drainage and water runoff from agricultural fields. The model was purely time dependent. To model the scenario more realistically, we study here a reaction-diffusion system, taking into account diffusion in two spatial dimensions. Concentration of nutrients, density of algae and density of detritus are considered as dynamical variables. Model equations are analyzed using the theory of partial differential equations and dynamical systems. The analytically obtained results show that self-diffusion cannot alter the local asymptotical stability of the uniform steady state but the unstable uniform steady state can be made stable by increasing diffusion coefficients in the case of global stability. The modified model cannot explain the patterns of algal bloom in Turing's sense. We modify the model with taking into account cross-diffusion. We observe that this newly modified model can explain different kinds of patterns of algal blooms. [ABSTRACT FROM AUTHOR]

With the foremost spotlight on the host-pathogen connections, research in the field of infected population and its subsequent recovery is intended for mounting how the immune structure defends against the infectious diseases. Host-Pathogen interactions are the mathematical prototypes pertaining to epidemiology and these are of enormous importance in view of the appearance and recurrence of epidemiological diseases in the days to come of global perspective. In this research article we extend the work of [1] and incorporate the fractional-order differential equations into the mathematical model of host-pathogen interactions with effect of recovery for infected population to study the effect of it on the system dynamics. Analytical study on the basis of stability analysis with fractional derivative is observed. Furthermore, numerical illustration through FDE12 method is furnished. It is an implementation of the predictor-corrector method of Adams-Bashforth-Moulton which has been applied for solving the fractional-order differential equations to support the analytical results. [ABSTRACT FROM AUTHOR]

Investigating disease progression, transmission of infection and impacts of Multidrug Therapy (MDT) to inhibit demyelination in leprosy involves a certain amount of difficulty in terms of the in-built uncertain complicated and complex intracellular cell dynamical interactions. To tackle this scenario and to elucidate a more realistic, rationalistic approach of examining the infection mechanism and associated drug therapeutic interventions, we propose a four-dimensional ordinary differential equation-based model. Stochastic processes has been employed on this deterministic system by formulating the Kolmogorov forward equation introducing a transition state and the quasi-stationary distribution, exact distribution analysis have been investigated which allow us to estimate an expected time to extinction of the infected Schwann cells into the human body more prominently. Additionally, to explore the impact of uncertainty in the key intracellular factors, the stochastic system is investigated incorporating random perturbations and environmental noises in the disease dissemination, proliferation and reinfection rates. Rigorous numerical simulations validating the analytical outcomes provide us significant novel insights on the progression of leprosy and unravelling the existing major treatment complexities. Analytical experiments along with the simulations utilizing Monte-Carlo method and Euler-Maruyama scheme involving stochasticity predicts that the bacterial density is underestimated due to the recurrence of infection and suggests that maintaining a drug-efficacy rate in the range 0.6-0.8 would be substantially efficacious in eradicating leprosy., Competing Interests: Declaration of competing interest The authors declare no conflict of interest at all., (Copyright © 2024 Elsevier Inc. All rights reserved.)

Several reports in India indicate hospitals and quarantined centers are COVID-19 hotspots. To study the transmission occurring from the hospitals and as well as from the community, we developed a mechanistic model with a lockdown effect. Using daily COVID-19 cases data from six states and overall India, we estimated several important parameters of our model. Moreover, we provided an estimation of the effective (R T ), the basic (R 0 ), the community (R C ), and the hospital (R H ) reproduction numbers. We forecast COVID-19 notified cases from May 3, 2020, till May 20, 2020, under five different lockdown scenarios in the seven locations. Our analysis suggests that 65% to 99% of the new COVID-19 cases are currently asymptomatic in those locations. Besides, about 1-16% of the total COVID-19 transmission are currently occurring from hospital-based contact and these percentage can increase up to 69% in some locations. Furthermore, the hospital-based transmission rate (β 2 ) has significant positive (0.65 to 0.8) and negative (-0.58 to -0.23) correlation with R 0 and the effectiveness of lockdown, respectively. Therefore, a much larger COVID-19 outbreak may trigger from the hospital-based transmission. In most of the locations, model forecast from May 3, 2020, till May 20, 2020, indicates a two-times increase in cumulative cases in comparison to total observed cases up to April 29, 2020. Based on our results, we proposed a containment policy that may reduce the threat of a larger COVID-19 outbreak in the future., (© 2021 Society for Risk Analysis.)

We considered a non-linear predator-prey model with an Allee effect on both populations on a two spatial dimension reaction-diffusion setup. Special importance to predator mortality was given as it may be often controlled through human-made harvesting processes. The local dynamics of the model was studied through boundedness, equilibrium, and stability analysis. An extensive numerical stability analysis was performed and found that bi-stability is not possible for the non-spatial model. By analyzing the spatial model, we found the condition for successful invasion and the persistence region of the species based on the predator Allee effect and its mortality parameter. Four different dynamics in this region of the parameter space are mainly explored. First, the Allee effect on both populations leads to various new types of species spread. Second, for a high value of per-capita growth rate, two completely new spreads (e.g., sun surface, colonial) have been found depending on the Allee effect parameter. Third, the Allee coefficient on the predator population leads to spatiotemporal chaos via a patchy spread for both linear and quadratic mortality rates. Finally, a more rigorous analysis is performed to study the chaotic nature of the system within the whole persistence domain. We have studied the possibility of chaos through temporal variation in different invasion regions. Furthermore, the chaotic fluctuation is studied through the sensitivity of initial conditions and by investigating the dominant Lyapunov exponent value.