Transcritical Bifurcation Research Articles

Mathematical Modelling and Optimal Control of Malaria Using Awareness-Based Interventions

Malaria is a serious illness caused by a parasite, called Plasmodium, transmitted to humans through the bites of female Anopheles mosquitoes. The parasite infects and destroys the red blood cells in the human body leading to symptoms, such as fever, headache, and flu-like illness. Awareness campaigns that educate people about malaria prevention and control reduce transmission of the disease. In this research, a mathematical model is proposed to study the impact of awareness-based control measures on the transmission dynamics of malaria. Some basic properties of the proposed model, such as non-negativity and boundedness of the solutions, the existence of the equilibrium points, and their stability properties, have been studied using qualitative theory. Disease-free equilibrium is globally asymptotic when the basic reproduction number, R0, is less than the number of current cases. Finally, optimal control theory is applied to minimize the cost of disease control and solve the optimal control problem by applying Pontryagin’s minimum principle. Numerical simulations have been provided for the confirmation of the analytical results. Endemic equilibrium exists for R0>1, and a forward transcritical bifurcation occurs at R0=1. The optimal profiles of the treatment process, organizing awareness campaigns, and insecticide uses are obtained for the cost-effectiveness of malaria management. This research concludes that awareness campaigns through social media with an optimal control approach are best for cost-effective malaria management.

Persistence and bifurcation analysis among four species interactions with the influence of competition, predation and harvesting

In this paper, the interplay among four population species is offered. The system consists of two competitive prey, predator and super predators. The application of the hypothesis of the Sotomayor theorem for local bifurcation around every equilibrium point is adopted. It is detected that the transcritical bifurcation could occur near most of the system's equilibrium points, while saddle-node and pitchfork bifurcation can not be accrued at any of them. Further, the conditions that guarantee the accruing Hopf bifurcation are carried out. Finally, some numerical analysis is illustrated to confirm the analytical results.

A mathematical model for transmission dynamics of COVID-19 infection.

In this paper, a mathematical model of COVID-19 has been proposed to study the transmission dynamics of infection by taking into account the role of symptomatic and asymptomatic infected individuals. The model has also considered the effect of non-pharmaceutical interventions (NPIs) in controlling the spread of virus. The basic reproduction number ( ) has been computed and the analysis shows that for , the disease-free state becomes globally stable. The conditions of existence and stability for two other equilibrium states have been obtained. Transcritical bifurcation occurs when basic reproduction number is one (i.e. ). It is found that when asymptomatic cases get increased, infection will persist in the population. However, when symptomatic cases get increased as compared to asymptomatic ones, the endemic state will become unstable and infection may eradicate from the population. Increasing NPIs decrease the basic reproduction number and hence, the epidemic can be controlled. As the COVID-19 transmission is subject to environmental fluctuations, the effect of white noise has been considered in the deterministic model. The stochastic differential equation model has been solved numerically by using the Euler-Maruyama method. The stochastic model gives large fluctuations around the respective deterministic solutions. The model has been fitted by using the COVID-19 data of three waves of India. A good match is obtained between the actual data and the predicted trajectories of the model in all three waves of COVID-19. The findings of this model may assist policymakers and healthcare professionals in implementing the most effective measures to prevent the transmission of COVID-19 in different settings.

Modeling the Effect of TV and Social Media Advertisements on the Dynamics of Vector-Borne Disease Malaria

Vector-borne disease malaria is transmitted to humans by arthropod vectors (mosquitoes) and contributes significantly to the global disease burden. TV and social media play a key role to disseminate awareness among people by broadcasting awareness programs. In this paper, a nonlinear model is formulated and analyzed in which cumulative number of advertisements through TV and social media is taken as dynamical variable that propagates awareness among people to control the prevalence of vector-borne disease. The human population is partitioned into susceptible, infected and aware classes, while the vector population is divided into susceptible and infected classes. Humans become infected and new cases arise when bitten by infected vectors (mosquitoes) and susceptible vectors get infected as they bite infected humans. The feasibility of equilibria is justified and their stability conditions are discussed. A crucial parameter, basic reproduction number, which measures the disease transmission potentiality is obtained. Bifurcation analysis is performed by varying the sensitive parameters, and it is found that the proposed system shows different kinds of bifurcations, such as transcritical bifurcation, saddle-node bifurcation and Hopf bifurcation, etc. The analysis of the model shows that reduction in vector population due to intervention of people of aware class would not efficiently reduce the infective cases, rather we have to minimize the transmission rates anyhow, to control the disease outbreak.

Numerical Analysis of Ordinary Differential Equations of Ecological Competing Species Across Diverse Environments

In a geographical region, species have their range margins (i.e., the geographic boundaries where species can be found). Several species distribution models have shown that environmental factors (i.e., abiotic factors) and species interactions (i.e., biotic interactions) are responsible for shaping the distributions of species. Yet, most of the models often focus on one of these factors and ignore their joint effects. Consequently, predicting which species will exist and at what range margins is a challenge in ecology. Thus, in this paper, the combined influences of these ecological factors on multi-species community structures are studied. An ordinary differential equations (ODE) model is employed to study multi-species competition interactions across diverse environments. The model is numerically analysed for the range margins of the species and threshold values of competition strength which leads to the presence-absence of species. It is observed that the range margins are influenced by competition between species combined with environmental factors and the threshold values of competition strength correspond to transcritical bifurcation. Depending on the species’ competition strengths, the model exhibits coexistence and exclusion of species, mediated by weak and aggressive biotic interactions, respectively. It is observed that ecologically similar species competitively affect each other more than dissimilar species.

Stability and bifurcations of an SIR model with a nonlinear incidence rate

In this paper, an SIR model with a nonlinear incidence rate is studied. A disease‐free equilibrium , an endemic equilibrium , and the basic reproduction number of the model are obtained. If is locally asymptotically stable and if is locally asymptotically stable. By Barbalat's lemma, we study the global stability of the model. Transcritical bifurcation analysis is investigated by using the Sotomayor theorem. As the infection rate increases, the asymptotic behavior of the system near approaches and the system has a transcritical bifurcation. Also, we check the existence of Hopf bifurcation for the given system. In addition, a sensitivity analysis is provided for the basic reproduction number. Our results are supported with numerical simulations.

Bifurcation theory of attractors and minimal sets in d-concave nonautonomous scalar ordinary differential equations

Two one-parametric bifurcation problems for scalar nonautonomous ordinary differential equations are analyzed assuming the coercivity of the time-dependent function determining the equation and the concavity of its derivative with respect to the state variable. The skewproduct formalism leads to the analysis of the number and properties of the minimal sets and of the shape of the global attractor, whose abrupt variations determine the occurrence of local saddle-node, local transcritical and global pitchfork bifurcation points of minimal sets and of discontinuity points of the global attractor.

A Dynamic Programming Approach to Ecosystem Management

We propose a way of dealing with invasive species or pest control in agriculture. Ecosystems can be modeled via dynamical systems. For their study, it is necessary to establish their possible equilibria. Even a moderately complex system exhibits, in general, multiple steady states. Usually, they are related to each other through transcritical bifurcations, i.e., the system settles to a different equilibrium when some bifurcation parameter crosses a critical threshold. From a situation in which the pest is endemic, it is desirable to move to a pest-free point. The map of the system’s equilibria and their connections via transcritical bifurcations may indicate a path to attain the desired state. However, to force the parameters to cross the critical threshold, some human action is required, and this effort has a cost. The tools of dynamic programming allow the detection of the cheapest path to reach the desired goal. In this paper, an algorithm for the solution to this problem is illustrated.

Fractional-order crime propagation model with non-linear transmission rate

Various studies present different mathematical models of ordinary and fractional differential equations to reduce delinquent behavior and encourage prosocial growth. However, these models do not consider the non-linear transmission rate, which depicts reality better than the linear transmission rate, as the relationship between non-criminals and criminals is not linear. In light of this, a novel fractional-order mathematical crime propagation model with a non-linear Beddington–DeAngelis transmission rate is proposed that divides the entire population into three clusters. The present study also compares the crime transmission models for various transmission rates, followed by an analytical investigation. The model shows two equilibrium points (criminal-free and crime-persistence equilibrium). The criminal-free equilibrium is locally and globally asymptotically stable when the criminal generation number is less than one. The crime-persistence equilibrium point does not appear until the criminal generation number exceeds one. In addition, this research investigates the incidence of transcritical bifurcation at the criminal-free equilibrium point. Furthermore, numerical simulations are performed to demonstrate the analytical results. In summary, the finding of this research suggests that as the order of derivative increases, the population approaches equilibrium more swiftly, and criminals decline with time for the different order of derivative.

Combining impact of velocity, fear and refuge for the predator–prey dynamics

We develop a deterministic predator–prey compartmental model to investigate the impact of their velocities on their interactions. Prey hides in a refuge area and comes out of this area when predation pressure declines. To avoid predation, prey can limit their velocity. For antipredator behaviour, we examined that prey mortality increases when either predator or prey velocity increases while raising antipredator behaviour increases prey density. We proved that predator free equilibrium is globally asymptotically stable and co-existing equilibrium will be globally stable under certain conditions. We find that transcritical bifurcations occur at predator-free equilibrium at the certain value of the death rate of the predator.

Disease dynamics and optimal control strategies of a two serotypes dengue model with co-infection

This paper aims to explore stability and bifurcations with control analysis of a co-infected two serotypes dengue model in presence of three controls, namely protection control, treatment control and mosquito killing efforts. First we analyze the model incorporating with constant controls then find the control path considering variable control. We examine biological feasibility of the considered model along with existence and stability criteria of different equilibrium with respect to basic reproduction number. Stability of serotype-I, serotype-II free equilibrium and positive co-existence equilibrium are investigated. Center manifold theorem is used to prove the stability of the co-infected endemic equilibrium. The model experiences Transcritical bifurcation with respect to basic reproduction number. Sensitivity analysis has been employed to identify most influential model parameters to control the infection. The time dependent optimal control problem is solved analytically and numerically using Pontryagin’s maximum principle. At last efficiency analysis has been carried out to find out more suitable control to combat the dengue disease. It is established from the analysis, protection control with treatment is more powerful than mosquito killing efforts by humans with treatment for controlling dengue.

Toxicity-mediated regime shifts in a contaminated nutrient-plankton system.

In this article, we contemplate the dynamics of a three-tier system of nutrient, phytoplankton, and zooplankton with a gestation delay of discrete type and a distributed delay in nutrient recycling. Phytoplankton secretion-mediated alteration in the grazing pattern of zooplankton is encapsulated by a Monod-Haldane functional response. We carry out global sensitivity analysis for identifying the crucial model parameters having a significant impact on zooplankton density. The system potentially exhibits bistable configurations under identical ecological conditions by allowing different bifurcation scenarios, including multiple saddle-node and transcritical bifurcations with varying input rates of nutrients and inhibitory effects of phytoplankton against zooplankton. We observe that the gestation delay in zooplankton is responsible for the emergence of noxious bloom events. Interestingly, when the delay parameter crosses a threshold, the system experiences chaotic disorder, which prognosticates the onset of irregular bloom. Furthermore, by adding Gaussian white noise, we have extended the deterministic model to its stochastic counterpart. We found that white noise appears to regulate the survival and extinction of interacting populations. Comprehensive numerical simulations are consistent with mathematical results prognosticated by linear analysis.

Modeling the Spread and Control of Viral Infection in Damaged Aquatic System: Emergence of Patterns

In this paper, we propose an eco-epidemiological model for prey (free-living motile species, i.e., dinoflagellates) and predator in an aquatic system. The predator’s density is affected by both virus-induced prey and toxicity. The spread of viral infection is taken as simple law of mass, and interaction between prey and predator is described by the Holling type II functional response. Stability analysis of the model system has been studied for spatial and non-spatial systems, and the theoretical results are verified with numerical simulation. It is explored that the aquatic system is very sensitive to carrying capacity and able to generate chaotic phenomena. We have observed the Hopf and transcritical bifurcation scenario for different carrying capacity and transmission of virus contamination. The results show that high carrying capacity affects the aquatic system, and virus also influences the species dynamics. Further, the diffusion-driven Turing instability is analyzed and different hot and cold cluster-based Turing patterns are observed, emphasizing the effect of time variation and virus contamination. The results obtained in this study give a rich dynamics and show that the viruses with poor water quality, high nutrients, red and green algae, toxic substances, etc. are responsible for natural substances demises, aquatic habitats death. In large, it affects the environment and human health.

Degenerate T-singularity bifurcation and crossing periodic orbits in a 3-dimensional piecewise smooth system

In this paper we investigate the dynamics of a 3-dimensional piecewise smooth system, which is an unfolding of a degenerate T-singularity. In local aspect, under the same non-degenerate condition given in [6] and determined by the linear part of perturbation, we analyze the bifurcation of this degenerate T-singularity on the switching manifold and give the bifurcation diagram including the degenerate two-fold bifurcation curve and the transcritical bifurcation curve. When this condition is not satisfied, a further non-degenerate condition determined by terms no more than degree two of perturbation is constructed, under which we obtain the pseudo-equilibrium bifurcation curve besides previous two bifurcation curves. In non-local aspect, for a system satisfying the second non-degenerate condition we prove that there exist either one family or two families of non-isolated crossing periodic orbits or at most two crossing periodic orbits and it is reachable. Corresponding necessary and sufficient conditions are given as well as periods and locations of these crossing periodic orbits.

Bifurcation analysis of an SIR model considering hospital resources and vaccination

In this paper, an SIR epidemic model considering hospital resources and vaccination is established and the rich dynamics and complex bifurcations are investigated. Firstly, the existence of disease-free equilibrium and endemic equilibria is explored. It is founded that when the vaccination rate is not high, the number of endemic equilibrium points changed easily with the number of hospital resources and vaccination, resulting in transcritical bifurcation and saddle–node bifurcations. Secondly, different types singularities such as degenerate saddle–node of codimension 1 at the disease-free equilibrium, and cusp or focus type Bogdanov–Takens singularities of codimension 3 at endemic equilibria are presented. Thirdly, bifurcation analysis at these equilibria is investigated, and it is found that the system undergoes a sequence of bifurcations, including Hopf, degenerate Hopf bifurcation, homoclinic bifurcation, the cusp type Bogdanov–Takens bifurcation of codimension 2, and the focus type Bogdanov–Takens bifurcation of codimension 3 which are the organizing centers for a series of bifurcations with lower codimension. And the system shows very rich dynamics such as the existence of multiple coexistent periodic orbits, homoclinic loops. Finally, numerical simulations are presented to verify the theoretical results.

Stoichiometry and environmental change drive dynamical complexity and unpredictable switches in an intraguild predation model

We incorporate stoichiometry (the balance of key elements) into an intraguild predation (IGP) model. Theoretical and numerical results show that our system exhibits complex dynamics, including chaos and multiple types of both bifurcations and bistability. Types of bifurcation present include saddle-node, Hopf, and transcritical bifurcations, and types of bistability present include node-node, node-cycle, and cycle-cycle bistability; cycle-cycle bistability has never been observed in IGP ordinary differential equation models. Stoichiometry can stabilize or destabilize the system via the disappearance or appearance of chaos. The species represented in the model can coexist for moderate levels of light intensity and nutrient availability. When the amount of light or nutrients present is extremely high or low, coexistence of the species becomes impossible, potentially harming biodiversity. Interestingly, stoichiometry can facilitate the re-emergence of severely endangered species as light intensity increases. In a temporally changing environment, the system can jump between different unstable states following changes in light intensity, with the trajectory followed depending strongly on initial conditions.

A Study of a Prey-Generalist Predator System Considering Hunting Cooperation and Fear Effects Under Interval Uncertainty

In this paper, we examine the role of fear and hunting cooperation in a generalist predator–prey system in an imprecise environment. We investigated the positivity and boundedness of the solutions and determined the existence of equilibria and their local stability for the system. We analyzed the transcritical and Hopf bifurcation considering the uncertain parameter as a bifurcation parameter. Based on simulation results, both populations can survive at low to moderate values of imprecise parameters, but the prey population cannot survive at higher values of the imprecise parameter. Additionally, we observe fear, hunting cooperation, and generalist predators with an imprecise concept that illustrates the complex dynamics of the system.

Dynamic complexity of a modified Leslie–Gower predator–prey system with fear effect

In this paper, the influence of the fear effect and Leslie–Gower function on the dynamic behavior of the predator–prey model is considered. First, the well-posedness analysis of this model is demonstrated, and the existence and local stability of the equilibrium point are given. Then by using the bifurcation theory and selecting the appropriate bifurcation parameters, many types of bifurcation phenomena in the system are discovered, including transcritical bifurcation, Hopf bifurcation and Bogdanov–Takens bifurcation. Meanwhile, the numerical simulations on the basis of theoretical analysis are further carried out to intuitively show the influence of fear effect on population. As the degree of fear increases, the system will undergo multiple dynamic behaviors switching until the final extinction of prey population, while the predator population will survive due to the presence of substitute prey. In addition, the initial density of predator and prey can determine whether the solution of system tends to a coexisting steady state or a periodic oscillation.

Transmission dynamics of monkeypox virus with treatment and vaccination controls: a fractional order mathematical approach

Presently, monkeypox virus infection has spread worldwide in the ongoing outbreak that began in the UK. To study the transmission dynamics of monkeypox, we formulate here a seven-compartmental (five compartments for the human population and two compartments for animals or rodents) fractional-order mathematical model. The existence and uniqueness of the solution of the proposed fractional order model are examined here. The basic reproduction number for humans () and animals () are obtained through the next-generation matrix approach. Depending on the values of and , we observed that the fractional order model has three equilibria, namely, monkeypox-free equilibrium, animal-free endemic equilibrium, and endemic equilibrium. Also, the stability of all equilibria is checked in this present article. We found that the model goes through transcritical bifurcation at for any values of and at for . Best of our knowledge, this is the first work where the fractional order optimal control for monkeypox is formulated and solved considering vaccination and treatment controls. Several feasible parameter values are used in the simulations to visualize and verify the findings, from which the results show that fractional order is more appropriate. Finally, parameters involved in the expression of and are scaled using the sensitivity index approach.

More complex dynamics in a discrete prey-predator model with the Allee effect in prey.

In this paper, we revisit a discrete prey-predator model with the Allee effect in prey to find its more complex dynamical properties. After pointing out and correcting those known errors for the local stability of the unique positive fixed point $ E_*, $ unlike previous studies in which the author only considered the codim 1 Neimark-Sacker bifurcation at the fixed point $ E_*, $ we focus on deriving many new bifurcation results, namely, the codim 1 transcritical bifurcation at the trivial fixed point $ E_1, $ the codim 1 transcritical and period-doubling bifurcations at the boundary fixed point $ E_2, $ the codim 1 period-doubling bifurcation and the codim 2 1:2 resonance bifurcation at the positive fixed point $ E_* $. The obtained theoretical results are also further illustrated via numerical simulations. Some new dynamics are numerically found. Our new results clearly demonstrate that the occurrence of 1:2 resonance bifurcation confirms that this system is strongly unstable, indicating that the predator and the prey will increase rapidly and breakout suddenly.