Computational dynamics of a Lotka-Volterra Model with additive Allee effect based on Atangana-Baleanu fractional derivative

This paper studies an interaction between one prey and one predator following Lotka-Volterra model with additive Allee effect in predator. The Atangana-Baleanu fractional-order derivative is used for the operator. Since the theoretical ways to investigate the model using this operator are limited, the dynamical behaviors are identified numerically. By simulations, the influence of the order of the derivative on the dynamical behaviors is given. The numerical results show that the order of the derivative may impact the convergence rate, the occurrence of Hopf bifurcation, and the evolution of the diameter of the limit-cycle.

Mathematical model with fractional order derivatives for Tuberculosis taking into account its relationship with HIV/AIDS and Diabetes

In this paper, we present a mathematical model for the study of resistance to tuberculosis treatment using fractional derivatives in the Caputo sense. This model takes into account the relationship between Tuberculosis, HIV/AIDS, and diabetes and differentiates resistance cases into MDR-TB (multidrug-resistant tuberculosis) and XDR-TB (extensively drug-resistant tuberculosis). We present the basic results associated with the model and study the behavior of the disease-free equilibrium points in the different sub-populations, TB-Only, TB-HIV/AIDS, and TB-Diabetes. We performed computational simulations for different fractional orders (α-values) using an Adams-Bashforth-Moulton type predictor-corrector PECE method. Among the results obtained, we have that the MDR-TB cases in all sub-populations decrease at the beginning of the study for the different α-values. In XDR-TB cases in the TB-Only sub-population, there is a decrease in the number of cases. XDR-TB cases in the TB-HIV/AIDS sub-population have differentiated behavior depending on α. This knowledge helps to design an effective control strategy. The XDR-TB cases in diabetics increased throughout the study period and outperformed all resistant compartments for the different α-values. We recommend special attention to the control of this compartment due to this growth.

Model matematika penyebaran COVID-19 dengan penggunaan masker kesehatan dan karantina

This study developed a model for the spread of COVID-19 disease using the SIR model which was added by a health mask and quarantine for infected individuals. The population is divided into six subpopulations, namely the subpopulation susceptible without a health mask, susceptible using a health mask, infected without using a health mask, infected using a health mask, quarantine for infected individuals, and the subpopulation to recover. The results obtained two equilibrium points, namely the disease-free equilibrium point and the endemic equilibrium point, and the basic reproduction number (R0). The existence of a disease-free equilibrium point is unconditional, whereas an endemic equilibrium point exists if the basic reproduction number is more than one. Stability analysis of the local asymptotically stable disease-free equilibrium point when the basic reproduction number is less than one. Furthermore, numerical simulations are carried out to provide a geometric picture related to the results that have been analyzed. The results of numerical simulations support the results of the analysis obtained. Finally, the sensitivity analysis of the basic reproduction numbers carried out obtained four parameters that dominantly affect the basic reproduction number, namely the rate of contact of susceptible individuals with infection, the rate of health mask use, the rate of health mask release, and the rate of quarantine for infected individuals.

Dynamics in two competing predators-one prey system with two types of Holling and fear effect

In this article, it is formulated a predator-prey model of two predators consuming a single limited prey resource. On the other hand, two predators have to compete with each other for survival. The predation function for two predators is assumed to be different where one predator uses Holling type I while the other uses Holling type II. It is also assumed that the fear effect is considered in this model as indirect influence evoked by both predators. Non-negativity and boundedness is written to show the biological justification of the model. Here, it is found that the model has five equilibrium points existed under certain condition. We also perform the local stability analysis on the equilibrium points with three equilibrium points are stable under certain conditions and two equilibrium points are unstable. Hopf bifurcation is obtained by choosing the consumption rate of the second predator as the bifurcation parameter. In the last part, several numerical solutions are given to support the analysis results.

Fear effect in discrete prey-predator model incorporating square root functional response

In this work, an interaction between prey and its predator involving the effect of fear in presence of the predator and the square root functional response is investigated. Fixed points and their stability condition are calculated. The conditions for the occurrence of some phenomena namely Neimark-Sacker, Flip, and Fold bifurcations are given. Base on some hypothetical data, the numerical simulations consist of phase portraits and bifurcation diagrams are demonstrated to picturise the dynamical behavior. It is also shown numerically that rich dynamics are obtained by the discrete model as the effect of fear.

Dynamics of a stage–structure Rosenzweig–MacArthur model with linear harvesting in prey and cannibalism in predator

A kind of stage-structure Rosenzweig–MacArthur model with linear harvesting in prey and cannibalism in predator is investigated in this paper. By analyzing the model, local stability of all possible equilibrium points is discussed. Moreover, the model undergoes a Hopf–bifurcation around the interior equilibrium point. Numerical simulations are carried out to illustrate our main results.

Analisis kestabilan dan kontrol optimal model matematika penyebaran penyakit Ebola dengan variabel kontrol berupa karantina

Ebola disease is an infectious disease caused by a virus from the genus Ebolavirus and the family Filoviridae. Ebola disease is one of the most deadly diseases for human. The purpose of the thesis is to analyze the stability of the equilibrium point and to apply the optimal control of quarantine on a mathematical model of the spread of ebola. Model without control has two equilibria, non-endemic equilibrium and endemic equilibrium. The existence of endemic equilibrium and local stability depends on the basic reproduction number (R0). The non-endemic equilibrium is asymptotically stable if R0 1 and endemic equilibrium tend to asymptotically stable if R0 1. The problem of optimal control is solved by Pontryagin’s Maximum Principle. From the numerical simulation, the result shows that control is effective enough to minimize the number of infected human population and to minimize the cost of its control.

Analisis dinamik model predator-prey tipe Gause dengan wabah penyakit pada prey

This article studies the dynamics of a Gause-type predator-prey model with infectious disease in the prey. The constructed model is a deterministic model which assumes the prey is divided into two compartments i.e. susceptible prey and infected prey, and both of them are hunted by predator bilinearly. It is investigated that there exist five biological equilibrium points such as all population extinction point, infected prey and predator extinction point, infected prey extinction point, predator extinction point, and co-existence point. We find that all population extinction point always unstable while others are conditionally locally asymptotically stable. Numerical simulations, as well as the phase portraits, are given to support the analytical results.

Estimasi Reproduction Number Model Matematika Penyebaran Malaria di Sumba Tengah, Indonesia

Malaria is transmitted via a bite of mosquitoes and it is dangerous if it is not properly treated. Mathematical modeling can be formulated to understand the disease transmission dynamics. In this paper, a mathematical model with an awareness program has been formulated and the reproduction number has been estimated against the data from Weeluri Health Center, Mamboro District, Central Sumba. The calculation showed that the reproduction number is R0 = 1.2562. Results showed that if the efficacy of the awareness program is lower than 20%, the reproduction number is still above unity. If the efficacy of the awareness program is higher than 20%, the reproduction number is lower than unity. This implies that the efficacy of awareness programs is the key to the success of Malaria eradication.

Impact of predator fear on two competing prey species

Predator-prey interaction is a fundamental feature in the ecological system. The majority of studies have addressed how competition and predation affect species coexistence. Recent field studies on vertebrate has shown that fear of predators can influence the behavioural pattern of prey populations and reduce their reproduction. A natural question arises whether species coexistence is still possible or not when predator induce fear on competing species. Based on the above observation, we propose a mathematical model of two competing prey-one predator system with the cost of fear that affect not only the reproduction rate of both the prey population but also the predation rate of predator. To make the model more realistic, we incorporate intraspecific competition within the predator population. Biological justification of the model is shown through positivity and boundedness of solutions. Existence andstability of different boundary equilibria are discussed. Condition for the existence of coexistence equilibrium point is derived from showing uniform persistence. Local as well as a global stability criterion is developed. Bifurcation analysis is performed by choosing the fear effect as the bifurcation parameter of the model system. The nature of the limit cycle emerging through a Hopf bifurcation is indicated. Numerical experiments are carried out to test the theoretical results obtained from this model.