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34,354,966 active cases and 460,787 deaths because of COVID-19 pandemic were recorded on November 06, 2021, in India. To end this ongoing global COVID-19 pandemic, there is an urgent need to implement multiple population-wide policies like social distancing, testing more people and contact tracing. To predict the course of the pandemic and come up with a strategy to control it effectively, a compartmental model has been established. The following six stages of infection are taken into consideration: susceptible ( S ), asymptomatic infected ( A ), clinically ill or symptomatic infected ( I ), quarantine ( Q ), isolation ( J ) and recovered ( R ), collectively termed as SAIQJR. The qualitative behavior of the model and the stability of biologically realistic equilibrium points are investigated in terms of the basic reproduction number. We performed sensitivity analysis with respect to the basic reproduction number and obtained that the disease transmission rate has an impact in mitigating the spread of diseases. Moreover, considering the non-pharmaceutical and pharmaceutical intervention strategies as control functions, an optimal control problem is implemented to mitigate the disease fatality. To reduce the infected individuals and to minimize the cost of the controls, an objective functional has been constructed and solved with the aid of Pontryagin’s maximum principle. The implementation of optimal control strategy at the start of a pandemic tends to decrease the intensity of epidemic peaks, spreading the maximal impact of an epidemic over an extended time period. Extensive numerical simulations show that the implementation of intervention strategy has an impact in controlling the transmission dynamics of COVID-19 epidemic. Further, our numerical solutions exhibit that the combination of three controls are more influential when compared with the combination of two controls as well as single control. Therefore, the implementation of all the three control strategies may help to mitigate novel coronavirus disease transmission at this present epidemic scenario.

We report a mathematical model which depicts the spatiotemporal dynamics of glioma cells, macrophages, cytotoxic-T-lymphocytes, immuno-suppressive cytokine TGF-β and immuno-stimulatory cytokine IFN-γ through a system of five coupled reaction-diffusion equations. We performed local stability analysis of the biologically based mathematical model for the growth of glioma cell population and their environment. The presented stability analysis of the model system demonstrates that the temporally stable positive interior steady state remains stable under the small inhomogeneous spatiotemporal perturbations. The irregular spatiotemporal dynamics of gliomas, macrophages and cytotoxic T-lymphocytes are discussed extensively and some numerical simulations are presented. Performed some numerical simulations in both one and two dimensional spaces. The occurrence of heterogeneous pattern formation of the system has both biological and mathematical implications and the concepts of glioma cell progression and invasion are considered. Simulation of the model shows that by increasing the value of time, the glioma cell population, macrophages and cytotoxic-T-lymphocytes spread throughout the domain.

T11 Target structure (T11TS), a membrane glycoprotein isolated from sheep erythrocytes, reverses the immune suppressed state of brain tumor induced animals by boosting the functional status of the immune cells. This study aims at aiding in the design of more efficacious brain tumor therapies with T11 target structure. We propose a mathematical model for brain tumor (glioma) and the immune system interactions, which aims in designing efficacious brain tumor therapy. The model encompasses considerations of the interactive dynamics of glioma cells, macrophages, cytotoxic T-lymphocytes (CD8(+) T-cells), TGF-β, IFN-γ and the T11TS. The system undergoes sensitivity analysis, that determines which state variables are sensitive to the given parameters and the parameters are estimated from the published data. Computer simulations were used for model verification and validation, which highlight the importance of T11 target structure in brain tumor therapy.

Algal blooms are increasing in coastal waters worldwide. The study on the features of algal pollution in water bodies and the ways to eliminate them is of vital importance. Preventing, treating, and monitoring algal blooms can be an unanticipated cost for a water system. To tame algal bloom in a lake, the government provides funds through budget allocation. In this paper, we propose a mathematical model to investigate the effect of budget allocation on the control of algal bloom in a lake. We assume that the growth of budget follows logistic law and also increases in proportion to the algal density in the lake. A part of the budget is utilized for the control of inflow of nutrients, while the remaining is used in the removal of algae from the lake. Our results show that algal bloom can be mitigated from the lake by reducing the inflow rate of nutrients to a very low value, which can be achieved for very high efficacy of budget allocation for the control of nutrients inflow from outside sources. Also, increasing the efficacy of budget allocation for the removal of algae helps to control the algal bloom. Further, more budget should be used on the control of nutrient’s inflow than on the removal of algae, as the presence of nutrients in high concentration will immediately proliferate the growth of algae. Moreover, the combined effects of controlling the inflow of nutrients and removing algae at high rates will result in nutrients and algae-free aquatic environment. Further, we modify the model by considering a discrete time delay involved in the increment of budget due to increased density of algae in the lake. We observe that chaotic oscillations may arise via equilibrium destabilization on increasing the values of time delay. We apply basic tools of nonlinear dynamics such as Poincaré section and maximum Lyapunov exponent to confirm the chaotic behavior of the system.

Over the last 250 years, anthropogenic activity has increased atmospheric carbon dioxide by nearly 40%. This increase is mainly caused by human fossil fuel combustion and deforestation, which are the main causes of global warming. Phytoplankton of the world’s oceans synthesizes half of the carbon dioxide of the total Earth’s photosynthetic activity. Thus, phytoplankton plays a crucial role in controlling Earth’s climate. To study this scenario, we propose and analyze a mathematical model for the carbon-phytoplankton-zooplankton interaction dynamics. Positivity, boundedness, existence, and stability of biologically possible equilibrium points are studied. The system exhibits Hopf bifurcation with respect to the carbon capture coefficient and the criteria of Hopf bifurcation is established around the coexisting equilibrium. Complex spatiotemporal dynamics and patchy pattern formation are observed in the spatially explicit model. The proposed carbon-phytoplankton-zooplankton system incorporates the effect of global warming, and our simulation shows shifts in plankton seasonal dynamics.

In this study, we investigate a mathematical model that describes the interactive dynamics of a predator-prey system with different kinds of response function. The positivity, boundedness, and uniform persistence of the system are established. We investigate the biologically feasible singular points and their stability analysis. We perform a comparative study by considering different kinds of functional responses, which suggest that the dynamical behavior of the system remains unaltered, but the position of the bifurcation points altered. Our model system undergoes Hopf bifurcation with respect to the growth rate of the prey population, which indicates that a periodic solution occurs around a fixed point. Also, we observed that our predator-prey system experiences transcritical bifurcation for the prey population growth rate. By using normal form theory and center manifold theorem, we investigate the direction and stability of Hopf bifurcation. The biological implications of the analytical and numerical findings are also discussed in this study.

We propose and analyze a single-species population model subject to fear and its carry-over effect with the help of evolutionary game theory (EGT). We incorporate fear and carry-over cost in the growth of a single species resource population and the extensive analysis of our non-evolutionary model suggests that it can exhibit both weak and strong Allee effects. From the game theoretical viewpoint, we assume that the intrinsic growth rate r of the resource population and the attack rate a of the consumer population are functions of a mean phenotypic trait ( u ) of the resource, following a Normal distribution. Evolutionary stable strategies (ESS) are determined by using ESS maximum principle. Our study of ESS suggests that species extinction may be avoided as a result of evolution, though the extinct equilibrium can also be an ESS under certain conditions. The ratio of variation in the intrinsic growth rate and the attack rate plays a significant role in the ESS conditions of different equilibria as well as the global dynamics of our EGT model. Numerical simulations are performed to support our theoretical analysis.

Intraguild predation, the most fundamental mechanism in food webs with three species, is a mix of competition and predation in which two species that are participating in a predator–prey relationship are also competing for a common resource. We investigate an intraguild predation model by considering three population, namely shared resource, IG-prey and IG-predator by using a coupled system of ordinary differential equations and use Holling type II functional response between IG-prey and IG-predator. The dynamic properties of the proposed model are analyzed, including positivity, boundedness, uniform persistent, the existence and stability of the equilibria and Hopf bifurcation. We derive the sufficient conditions for the persistence and extinction of all feasible scenarios for the proposed model by providing a comprehensive view of their global stability. Our results reveal that the attacking rate and decay rate of the intraguild-predator can promote the formation of the stability of the model by transforming it from an unstable state to a stable state. The region of stability around interior equilibrium and around IG-predator free equilibrium has been obtained in terms of the attacking rate and decay rate of the intraguild-predator. The parameters interval for the coexistence of the three species has been determined. Our theoretical analysis and numerical illustrations demonstrate that intraguild predation model is prone to have coexistence of all the three populations.

We present a mathematical model that explores the progression of Alzheimer’s disease, with a particular focus on the involvement of disease-related proteins and astrocytes. Our model consists of a coupled system of differential equations that delineates the dynamics of amyloid beta plaques, amyloid beta protein, tau protein, and astrocytes. Amyloid beta plaques can be considered fibrils that depend on both the plaque size and time. We change our mathematical model to a temporal system by applying an integration operation with respect to the plaque size. Theoretical analysis including existence, uniqueness, positivity, and boundedness is performed in our model. We extend our mathematical model by adding two populations, namely prion protein and amyloid beta-prion complex. We characterize the system dynamics by locating biologically feasible steady states and their local stability analysis for both models. The characterization of the proposed model can help inform in advancing our understanding of the development of Alzheimer’s disease as well as its complicated dynamics. We investigate the global stability analysis around the interior equilibrium point by constructing a suitable Lyapunov function. We validate our theoretical analysis with the aid of extensive numerical illustrations.

To understand the dynamics of neurodegenerative disease, we consider a fractional-order yeast prion mathematical model. We characterize the system dynamics by locating the steady states and perform their local stability analysis both theoretically and numerically. We employ the homotopy perturbation method to determine approximate solutions for the proposed model. By varying the fractional orders, we generate plots of these solutions over time to examine the dynamics of the Sup35 monomer and Sup35 prion populations. Our simulations reveal that the dynamical behaviors of both populations can undergo alterations depending upon the chosen fractional orders and the value of the threshold parameter.